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doc published by
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Known
df_mend__df_sdrg__df_cytp__df_topsep__df_toplnd__df_rcl__df_he__ax_frege1__ax_frege2__ax_frege8__ax_frege28__ax_frege31__ax_frege41__ax_frege52a__ax_frege54a__ax_frege58a__ax_frege52c__ax_frege54c
:
∀ x0 : ο .
(
wceq
cmend
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
csb
(
co
(
cv
x1
)
(
cv
x1
)
clmhm
)
(
λ x2 .
cun
(
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x2
)
)
(
cop
(
cfv
cnx
cplusg
)
(
cmpt2
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
co
(
cv
x3
)
(
cv
x4
)
(
cof
(
cfv
(
cv
x1
)
cplusg
)
)
)
)
)
(
cop
(
cfv
cnx
cmulr
)
(
cmpt2
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
ccom
(
cv
x3
)
(
cv
x4
)
)
)
)
)
(
cpr
(
cop
(
cfv
cnx
csca
)
(
cfv
(
cv
x1
)
csca
)
)
(
cop
(
cfv
cnx
cvsca
)
(
cmpt2
(
λ x3 x4 .
cfv
(
cfv
(
cv
x1
)
csca
)
cbs
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
co
(
cxp
(
cfv
(
cv
x1
)
cbs
)
(
csn
(
cv
x3
)
)
)
(
cv
x4
)
(
cof
(
cfv
(
cv
x1
)
cvsca
)
)
)
)
)
)
)
)
)
⟶
wceq
csdrg
(
cmpt
(
λ x1 .
cdr
)
(
λ x1 .
crab
(
λ x2 .
wcel
(
co
(
cv
x1
)
(
cv
x2
)
cress
)
cdr
)
(
λ x2 .
cfv
(
cv
x1
)
csubrg
)
)
)
⟶
wceq
ccytp
(
cmpt
(
λ x1 .
cn
)
(
λ x1 .
co
(
cfv
(
cfv
ccnfld
cpl1
)
cmgp
)
(
cmpt
(
λ x2 .
cima
(
ccnv
(
cfv
(
co
(
cfv
ccnfld
cmgp
)
(
cdif
cc
(
csn
cc0
)
)
cress
)
cod
)
)
(
csn
(
cv
x1
)
)
)
(
λ x2 .
co
(
cfv
ccnfld
cv1
)
(
cfv
(
cv
x2
)
(
cfv
(
cfv
ccnfld
cpl1
)
cascl
)
)
(
cfv
(
cfv
ccnfld
cpl1
)
csg
)
)
)
cgsu
)
)
⟶
wceq
ctopsep
(
crab
(
λ x1 .
wrex
(
λ x2 .
wa
(
wbr
(
cv
x2
)
com
cdom
)
(
wceq
(
cfv
(
cv
x2
)
(
cfv
(
cv
x1
)
ccl
)
)
(
cuni
(
cv
x1
)
)
)
)
(
λ x2 .
cpw
(
cuni
(
cv
x1
)
)
)
)
(
λ x1 .
ctop
)
)
⟶
wceq
ctoplnd
(
crab
(
λ x1 .
wral
(
λ x2 .
wceq
(
cuni
(
cv
x1
)
)
(
cuni
(
cv
x2
)
)
⟶
wrex
(
λ x3 .
wa
(
wbr
(
cv
x3
)
com
cdom
)
(
wceq
(
cuni
(
cv
x1
)
)
(
cuni
(
cv
x3
)
)
)
)
(
λ x3 .
cpw
(
cv
x1
)
)
)
(
λ x2 .
cpw
(
cv
x1
)
)
)
(
λ x1 .
ctop
)
)
⟶
wceq
crcl
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cint
(
cab
(
λ x2 .
wa
(
wss
(
cv
x1
)
(
cv
x2
)
)
(
wss
(
cres
cid
(
cun
(
cdm
(
cv
x2
)
)
(
crn
(
cv
x2
)
)
)
)
(
cv
x2
)
)
)
)
)
)
⟶
(
∀ x1 x2 :
ι → ο
.
wb
(
whe
x1
x2
)
(
wss
(
cima
x2
x1
)
x1
)
)
⟶
(
∀ x1 x2 : ο .
x1
⟶
x2
⟶
x1
)
⟶
(
∀ x1 x2 x3 : ο .
(
x1
⟶
x2
⟶
x3
)
⟶
(
x1
⟶
x2
)
⟶
x1
⟶
x3
)
⟶
(
∀ x1 x2 x3 : ο .
(
x1
⟶
x2
⟶
x3
)
⟶
x2
⟶
x1
⟶
x3
)
⟶
(
∀ x1 x2 : ο .
(
x1
⟶
x2
)
⟶
wn
x2
⟶
wn
x1
)
⟶
(
∀ x1 : ο .
wn
(
wn
x1
)
⟶
x1
)
⟶
(
∀ x1 : ο .
x1
⟶
wn
(
wn
x1
)
)
⟶
(
∀ x1 x2 x3 x4 : ο .
wb
x1
x2
⟶
wif
x1
x4
x3
⟶
wif
x2
x4
x3
)
⟶
(
∀ x1 : ο .
wb
x1
x1
)
⟶
(
∀ x1 x2 x3 : ο .
wa
x2
x3
⟶
wif
x1
x2
x3
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 x3 :
ι →
ι → ο
.
∀ x4 .
wceq
(
x2
x4
)
(
x3
x4
)
⟶
wsbc
x1
(
x2
x4
)
⟶
wsbc
x1
(
x3
x4
)
)
⟶
(
∀ x1 :
ι → ο
.
wceq
x1
x1
)
⟶
x0
)
⟶
x0
Theorem
df_mend
:
wceq
cmend
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
csb
(
co
(
cv
x0
)
(
cv
x0
)
clmhm
)
(
λ x1 .
cun
(
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x1
)
)
(
cop
(
cfv
cnx
cplusg
)
(
cmpt2
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
co
(
cv
x2
)
(
cv
x3
)
(
cof
(
cfv
(
cv
x0
)
cplusg
)
)
)
)
)
(
cop
(
cfv
cnx
cmulr
)
(
cmpt2
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
ccom
(
cv
x2
)
(
cv
x3
)
)
)
)
)
(
cpr
(
cop
(
cfv
cnx
csca
)
(
cfv
(
cv
x0
)
csca
)
)
(
cop
(
cfv
cnx
cvsca
)
(
cmpt2
(
λ x2 x3 .
cfv
(
cfv
(
cv
x0
)
csca
)
cbs
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
co
(
cxp
(
cfv
(
cv
x0
)
cbs
)
(
csn
(
cv
x2
)
)
)
(
cv
x3
)
(
cof
(
cfv
(
cv
x0
)
cvsca
)
)
)
)
)
)
)
)
)
(proof)
Theorem
df_sdrg
:
wceq
csdrg
(
cmpt
(
λ x0 .
cdr
)
(
λ x0 .
crab
(
λ x1 .
wcel
(
co
(
cv
x0
)
(
cv
x1
)
cress
)
cdr
)
(
λ x1 .
cfv
(
cv
x0
)
csubrg
)
)
)
(proof)
Theorem
df_cytp
:
wceq
ccytp
(
cmpt
(
λ x0 .
cn
)
(
λ x0 .
co
(
cfv
(
cfv
ccnfld
cpl1
)
cmgp
)
(
cmpt
(
λ x1 .
cima
(
ccnv
(
cfv
(
co
(
cfv
ccnfld
cmgp
)
(
cdif
cc
(
csn
cc0
)
)
cress
)
cod
)
)
(
csn
(
cv
x0
)
)
)
(
λ x1 .
co
(
cfv
ccnfld
cv1
)
(
cfv
(
cv
x1
)
(
cfv
(
cfv
ccnfld
cpl1
)
cascl
)
)
(
cfv
(
cfv
ccnfld
cpl1
)
csg
)
)
)
cgsu
)
)
(proof)
Theorem
df_topsep
:
wceq
ctopsep
(
crab
(
λ x0 .
wrex
(
λ x1 .
wa
(
wbr
(
cv
x1
)
com
cdom
)
(
wceq
(
cfv
(
cv
x1
)
(
cfv
(
cv
x0
)
ccl
)
)
(
cuni
(
cv
x0
)
)
)
)
(
λ x1 .
cpw
(
cuni
(
cv
x0
)
)
)
)
(
λ x0 .
ctop
)
)
(proof)
Theorem
df_toplnd
:
wceq
ctoplnd
(
crab
(
λ x0 .
wral
(
λ x1 .
wceq
(
cuni
(
cv
x0
)
)
(
cuni
(
cv
x1
)
)
⟶
wrex
(
λ x2 .
wa
(
wbr
(
cv
x2
)
com
cdom
)
(
wceq
(
cuni
(
cv
x0
)
)
(
cuni
(
cv
x2
)
)
)
)
(
λ x2 .
cpw
(
cv
x0
)
)
)
(
λ x1 .
cpw
(
cv
x0
)
)
)
(
λ x0 .
ctop
)
)
(proof)
Theorem
df_rcl
:
wceq
crcl
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cint
(
cab
(
λ x1 .
wa
(
wss
(
cv
x0
)
(
cv
x1
)
)
(
wss
(
cres
cid
(
cun
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
(
cv
x1
)
)
)
)
)
)
(proof)
Theorem
df_he
:
∀ x0 x1 :
ι → ο
.
wb
(
whe
x0
x1
)
(
wss
(
cima
x1
x0
)
x0
)
(proof)
Theorem
ax_frege1
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
x0
(proof)
Theorem
ax_frege2
:
∀ x0 x1 x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
(
x0
⟶
x1
)
⟶
x0
⟶
x2
(proof)
Theorem
ax_frege8
ax_frege8
:
∀ x0 x1 x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x1
⟶
x0
⟶
x2
(proof)
Theorem
ax_frege28
:
∀ x0 x1 : ο .
(
x0
⟶
x1
)
⟶
wn
x1
⟶
wn
x0
(proof)
Theorem
ax_frege31
:
∀ x0 : ο .
wn
(
wn
x0
)
⟶
x0
(proof)
Theorem
ax_frege41
:
∀ x0 : ο .
x0
⟶
wn
(
wn
x0
)
(proof)
Theorem
ax_frege52a
:
∀ x0 x1 x2 x3 : ο .
wb
x0
x1
⟶
wif
x0
x3
x2
⟶
wif
x1
x3
x2
(proof)
Theorem
ax_frege54a
:
∀ x0 : ο .
wb
x0
x0
(proof)
Theorem
ax_frege58a
:
∀ x0 x1 x2 : ο .
wa
x1
x2
⟶
wif
x0
x1
x2
(proof)
Theorem
ax_frege52c
ax_frege52c
:
∀ x0 :
ι → ο
.
∀ x1 x2 :
ι →
ι → ο
.
∀ x3 .
wceq
(
x1
x3
)
(
x2
x3
)
⟶
wsbc
x0
(
x1
x3
)
⟶
wsbc
x0
(
x2
x3
)
(proof)
Theorem
ax_frege54c
:
∀ x0 :
ι → ο
.
wceq
x0
x0
(proof)