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doc published by
PrCmT..
Known
df_coels__df_rels__df_ssr__df_refs__df_refrels__df_refrel__df_cnvrefs__df_cnvrefrels__df_cnvrefrel__df_syms__df_symrels__df_symrel__df_prt__ax_c5__ax_c4__ax_c7__ax_c10__ax_c10_b
:
∀ x0 : ο .
(
(
∀ x1 :
ι → ο
.
wceq
(
ccoels
x1
)
(
ccoss
(
cres
(
ccnv
cep
)
x1
)
)
)
⟶
wceq
crels
(
cpw
(
cxp
cvv
cvv
)
)
⟶
wceq
cssr
(
copab
(
λ x1 x2 .
wss
(
cv
x1
)
(
cv
x2
)
)
)
⟶
wceq
crefs
(
cab
(
λ x1 .
wbr
(
cin
cid
(
cxp
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
(
cin
(
cv
x1
)
(
cxp
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
cssr
)
)
⟶
wceq
crefrels
(
cin
crefs
crels
)
⟶
(
∀ x1 :
ι → ο
.
wb
(
wrefrel
x1
)
(
wa
(
wss
(
cin
cid
(
cxp
(
cdm
x1
)
(
crn
x1
)
)
)
(
cin
x1
(
cxp
(
cdm
x1
)
(
crn
x1
)
)
)
)
(
wrel
x1
)
)
)
⟶
wceq
ccnvrefs
(
cab
(
λ x1 .
wbr
(
cin
cid
(
cxp
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
(
cin
(
cv
x1
)
(
cxp
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
(
ccnv
cssr
)
)
)
⟶
wceq
ccnvrefrels
(
cin
ccnvrefs
crels
)
⟶
(
∀ x1 :
ι → ο
.
wb
(
wcnvrefrel
x1
)
(
wa
(
wss
(
cin
x1
(
cxp
(
cdm
x1
)
(
crn
x1
)
)
)
(
cin
cid
(
cxp
(
cdm
x1
)
(
crn
x1
)
)
)
)
(
wrel
x1
)
)
)
⟶
wceq
csyms
(
cab
(
λ x1 .
wbr
(
ccnv
(
cin
(
cv
x1
)
(
cxp
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
)
(
cin
(
cv
x1
)
(
cxp
(
cdm
(
cv
x1
)
)
(
crn
(
cv
x1
)
)
)
)
cssr
)
)
⟶
wceq
csymrels
(
cin
csyms
crels
)
⟶
(
∀ x1 :
ι → ο
.
wb
(
wsymrel
x1
)
(
wa
(
wss
(
ccnv
(
cin
x1
(
cxp
(
cdm
x1
)
(
crn
x1
)
)
)
)
(
cin
x1
(
cxp
(
cdm
x1
)
(
crn
x1
)
)
)
)
(
wrel
x1
)
)
)
⟶
(
∀ x1 :
ι → ο
.
wb
(
wprt
x1
)
(
wral
(
λ x2 .
wral
(
λ x3 .
wo
(
wceq
(
cv
x2
)
(
cv
x3
)
)
(
wceq
(
cin
(
cv
x2
)
(
cv
x3
)
)
c0
)
)
(
λ x3 .
x1
)
)
(
λ x2 .
x1
)
)
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 .
(
∀ x3 .
x1
x3
)
⟶
x1
x2
)
⟶
(
∀ x1 x2 :
ι → ο
.
(
∀ x3 .
(
∀ x4 .
x1
x4
)
⟶
x2
x3
)
⟶
(
∀ x3 .
x1
x3
)
⟶
∀ x3 .
x2
x3
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 .
wn
(
∀ x3 .
wn
(
∀ x4 .
x1
x4
)
)
⟶
x1
x2
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 x3 .
(
∀ x4 .
wceq
(
cv
x4
)
(
cv
x2
)
⟶
∀ x5 .
x1
x5
)
⟶
x1
x3
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 .
(
∀ x3 .
wceq
(
cv
x3
)
(
cv
x3
)
⟶
∀ x4 .
x1
x4
)
⟶
x1
x2
)
⟶
x0
)
⟶
x0
Theorem
df_coels
:
∀ x0 :
ι → ο
.
wceq
(
ccoels
x0
)
(
ccoss
(
cres
(
ccnv
cep
)
x0
)
)
(proof)
Theorem
df_rels
:
wceq
crels
(
cpw
(
cxp
cvv
cvv
)
)
(proof)
Theorem
df_ssr
:
wceq
cssr
(
copab
(
λ x0 x1 .
wss
(
cv
x0
)
(
cv
x1
)
)
)
(proof)
Theorem
df_refs
:
wceq
crefs
(
cab
(
λ x0 .
wbr
(
cin
cid
(
cxp
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
(
cin
(
cv
x0
)
(
cxp
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
cssr
)
)
(proof)
Theorem
df_refrels
:
wceq
crefrels
(
cin
crefs
crels
)
(proof)
Theorem
df_refrel
:
∀ x0 :
ι → ο
.
wb
(
wrefrel
x0
)
(
wa
(
wss
(
cin
cid
(
cxp
(
cdm
x0
)
(
crn
x0
)
)
)
(
cin
x0
(
cxp
(
cdm
x0
)
(
crn
x0
)
)
)
)
(
wrel
x0
)
)
(proof)
Theorem
df_cnvrefs
:
wceq
ccnvrefs
(
cab
(
λ x0 .
wbr
(
cin
cid
(
cxp
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
(
cin
(
cv
x0
)
(
cxp
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
(
ccnv
cssr
)
)
)
(proof)
Theorem
df_cnvrefrels
:
wceq
ccnvrefrels
(
cin
ccnvrefs
crels
)
(proof)
Theorem
df_cnvrefrel
:
∀ x0 :
ι → ο
.
wb
(
wcnvrefrel
x0
)
(
wa
(
wss
(
cin
x0
(
cxp
(
cdm
x0
)
(
crn
x0
)
)
)
(
cin
cid
(
cxp
(
cdm
x0
)
(
crn
x0
)
)
)
)
(
wrel
x0
)
)
(proof)
Theorem
df_syms
:
wceq
csyms
(
cab
(
λ x0 .
wbr
(
ccnv
(
cin
(
cv
x0
)
(
cxp
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
)
(
cin
(
cv
x0
)
(
cxp
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
cssr
)
)
(proof)
Theorem
df_symrels
:
wceq
csymrels
(
cin
csyms
crels
)
(proof)
Theorem
df_symrel
:
∀ x0 :
ι → ο
.
wb
(
wsymrel
x0
)
(
wa
(
wss
(
ccnv
(
cin
x0
(
cxp
(
cdm
x0
)
(
crn
x0
)
)
)
)
(
cin
x0
(
cxp
(
cdm
x0
)
(
crn
x0
)
)
)
)
(
wrel
x0
)
)
(proof)
Theorem
df_prt
:
∀ x0 :
ι → ο
.
wb
(
wprt
x0
)
(
wral
(
λ x1 .
wral
(
λ x2 .
wo
(
wceq
(
cv
x1
)
(
cv
x2
)
)
(
wceq
(
cin
(
cv
x1
)
(
cv
x2
)
)
c0
)
)
(
λ x2 .
x0
)
)
(
λ x1 .
x0
)
)
(proof)
Theorem
ax_c5
:
∀ x0 :
ι → ο
.
∀ x1 .
(
∀ x2 .
x0
x2
)
⟶
x0
x1
(proof)
Theorem
ax_c4
:
∀ x0 x1 :
ι → ο
.
(
∀ x2 .
(
∀ x3 .
x0
x3
)
⟶
x1
x2
)
⟶
(
∀ x2 .
x0
x2
)
⟶
∀ x2 .
x1
x2
(proof)
Theorem
ax_c7
:
∀ x0 :
ι → ο
.
∀ x1 .
wn
(
∀ x2 .
wn
(
∀ x3 .
x0
x3
)
)
⟶
x0
x1
(proof)
Theorem
ax_c10
:
∀ x0 :
ι → ο
.
∀ x1 x2 .
(
∀ x3 .
wceq
(
cv
x3
)
(
cv
x1
)
⟶
∀ x4 .
x0
x4
)
⟶
x0
x2
(proof)
Theorem
ax_c10_b
:
∀ x0 :
ι → ο
.
∀ x1 .
(
∀ x2 .
wceq
(
cv
x2
)
(
cv
x2
)
⟶
∀ x3 .
x0
x3
)
⟶
x0
x1
(proof)