Step |
Hyp |
Ref |
Expression |
1 |
|
dfon2 |
⊢ On = { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) } |
2 |
|
abeq1 |
⊢ ( { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) } = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ↔ ∀ 𝑥 ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ↔ 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ) ) |
3 |
|
vex |
⊢ 𝑥 ∈ V |
4 |
3
|
elrn |
⊢ ( 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ↔ ∃ 𝑦 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ) |
5 |
|
brin |
⊢ ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ↔ ( 𝑦 SSet 𝑥 ∧ 𝑦 ( Trans × V ) 𝑥 ) ) |
6 |
3
|
brsset |
⊢ ( 𝑦 SSet 𝑥 ↔ 𝑦 ⊆ 𝑥 ) |
7 |
|
brxp |
⊢ ( 𝑦 ( Trans × V ) 𝑥 ↔ ( 𝑦 ∈ Trans ∧ 𝑥 ∈ V ) ) |
8 |
3 7
|
mpbiran2 |
⊢ ( 𝑦 ( Trans × V ) 𝑥 ↔ 𝑦 ∈ Trans ) |
9 |
|
vex |
⊢ 𝑦 ∈ V |
10 |
9
|
eltrans |
⊢ ( 𝑦 ∈ Trans ↔ Tr 𝑦 ) |
11 |
8 10
|
bitri |
⊢ ( 𝑦 ( Trans × V ) 𝑥 ↔ Tr 𝑦 ) |
12 |
6 11
|
anbi12i |
⊢ ( ( 𝑦 SSet 𝑥 ∧ 𝑦 ( Trans × V ) 𝑥 ) ↔ ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ) |
13 |
5 12
|
bitri |
⊢ ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ↔ ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ) |
14 |
|
ioran |
⊢ ( ¬ ( 𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥 ) ↔ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥 ) ) |
15 |
|
brun |
⊢ ( 𝑦 ( I ∪ E ) 𝑥 ↔ ( 𝑦 I 𝑥 ∨ 𝑦 E 𝑥 ) ) |
16 |
3
|
ideq |
⊢ ( 𝑦 I 𝑥 ↔ 𝑦 = 𝑥 ) |
17 |
|
epel |
⊢ ( 𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥 ) |
18 |
16 17
|
orbi12i |
⊢ ( ( 𝑦 I 𝑥 ∨ 𝑦 E 𝑥 ) ↔ ( 𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥 ) ) |
19 |
15 18
|
bitri |
⊢ ( 𝑦 ( I ∪ E ) 𝑥 ↔ ( 𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥 ) ) |
20 |
14 19
|
xchnxbir |
⊢ ( ¬ 𝑦 ( I ∪ E ) 𝑥 ↔ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥 ) ) |
21 |
13 20
|
anbi12i |
⊢ ( ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ∧ ¬ 𝑦 ( I ∪ E ) 𝑥 ) ↔ ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥 ) ) ) |
22 |
|
brdif |
⊢ ( 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ∧ ¬ 𝑦 ( I ∪ E ) 𝑥 ) ) |
23 |
|
dfpss2 |
⊢ ( 𝑦 ⊊ 𝑥 ↔ ( 𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥 ) ) |
24 |
23
|
anbi1i |
⊢ ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ↔ ( ( 𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥 ) ∧ Tr 𝑦 ) ) |
25 |
|
an32 |
⊢ ( ( ( 𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥 ) ∧ Tr 𝑦 ) ↔ ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) ) |
26 |
24 25
|
bitri |
⊢ ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ↔ ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) ) |
27 |
26
|
anbi1i |
⊢ ( ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ↔ ( ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ) |
28 |
|
anass |
⊢ ( ( ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ↔ ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥 ) ) ) |
29 |
27 28
|
bitri |
⊢ ( ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ↔ ( ( 𝑦 ⊆ 𝑥 ∧ Tr 𝑦 ) ∧ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥 ) ) ) |
30 |
21 22 29
|
3bitr4i |
⊢ ( 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ) |
31 |
30
|
exbii |
⊢ ( ∃ 𝑦 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ∃ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ) |
32 |
|
exanali |
⊢ ( ∃ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 ∈ 𝑥 ) ↔ ¬ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ) |
33 |
31 32
|
bitri |
⊢ ( ∃ 𝑦 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ¬ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ) |
34 |
4 33
|
bitri |
⊢ ( 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ↔ ¬ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ) |
35 |
34
|
con2bii |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ↔ ¬ 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) |
36 |
|
eldif |
⊢ ( 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ↔ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ) |
37 |
3 36
|
mpbiran |
⊢ ( 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ↔ ¬ 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) |
38 |
35 37
|
bitr4i |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ↔ 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ) |
39 |
2 38
|
mpgbir |
⊢ { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) } = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) |
40 |
1 39
|
eqtri |
⊢ On = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) |