Step |
Hyp |
Ref |
Expression |
1 |
|
df-on |
⊢ On = { 𝑥 ∣ Ord 𝑥 } |
2 |
|
tz7.7 |
⊢ ( ( Ord 𝑥 ∧ Tr 𝑦 ) → ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥 ) ) ) |
3 |
|
df-pss |
⊢ ( 𝑦 ⊊ 𝑥 ↔ ( 𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥 ) ) |
4 |
2 3
|
bitr4di |
⊢ ( ( Ord 𝑥 ∧ Tr 𝑦 ) → ( 𝑦 ∈ 𝑥 ↔ 𝑦 ⊊ 𝑥 ) ) |
5 |
4
|
exbiri |
⊢ ( Ord 𝑥 → ( Tr 𝑦 → ( 𝑦 ⊊ 𝑥 → 𝑦 ∈ 𝑥 ) ) ) |
6 |
5
|
com23 |
⊢ ( Ord 𝑥 → ( 𝑦 ⊊ 𝑥 → ( Tr 𝑦 → 𝑦 ∈ 𝑥 ) ) ) |
7 |
6
|
impd |
⊢ ( Ord 𝑥 → ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ) |
8 |
7
|
alrimiv |
⊢ ( Ord 𝑥 → ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ) |
9 |
|
vex |
⊢ 𝑥 ∈ V |
10 |
|
dfon2lem3 |
⊢ ( 𝑥 ∈ V → ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → ( Tr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧 ) ) ) |
11 |
9 10
|
ax-mp |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → ( Tr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧 ) ) |
12 |
11
|
simpld |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → Tr 𝑥 ) |
13 |
9
|
dfon2lem7 |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → ( 𝑡 ∈ 𝑥 → ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ) ) |
14 |
13
|
ralrimiv |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ) |
15 |
|
dfon2lem9 |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → E Fr 𝑥 ) |
16 |
|
psseq2 |
⊢ ( 𝑡 = 𝑧 → ( 𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑧 ) ) |
17 |
16
|
anbi1d |
⊢ ( 𝑡 = 𝑧 → ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) ↔ ( 𝑢 ⊊ 𝑧 ∧ Tr 𝑢 ) ) ) |
18 |
|
elequ2 |
⊢ ( 𝑡 = 𝑧 → ( 𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑧 ) ) |
19 |
17 18
|
imbi12d |
⊢ ( 𝑡 = 𝑧 → ( ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ↔ ( ( 𝑢 ⊊ 𝑧 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑧 ) ) ) |
20 |
19
|
albidv |
⊢ ( 𝑡 = 𝑧 → ( ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ↔ ∀ 𝑢 ( ( 𝑢 ⊊ 𝑧 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑧 ) ) ) |
21 |
|
psseq1 |
⊢ ( 𝑢 = 𝑣 → ( 𝑢 ⊊ 𝑧 ↔ 𝑣 ⊊ 𝑧 ) ) |
22 |
|
treq |
⊢ ( 𝑢 = 𝑣 → ( Tr 𝑢 ↔ Tr 𝑣 ) ) |
23 |
21 22
|
anbi12d |
⊢ ( 𝑢 = 𝑣 → ( ( 𝑢 ⊊ 𝑧 ∧ Tr 𝑢 ) ↔ ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) ) ) |
24 |
|
elequ1 |
⊢ ( 𝑢 = 𝑣 → ( 𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧 ) ) |
25 |
23 24
|
imbi12d |
⊢ ( 𝑢 = 𝑣 → ( ( ( 𝑢 ⊊ 𝑧 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑧 ) ↔ ( ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) → 𝑣 ∈ 𝑧 ) ) ) |
26 |
25
|
cbvalvw |
⊢ ( ∀ 𝑢 ( ( 𝑢 ⊊ 𝑧 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑧 ) ↔ ∀ 𝑣 ( ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) → 𝑣 ∈ 𝑧 ) ) |
27 |
20 26
|
bitrdi |
⊢ ( 𝑡 = 𝑧 → ( ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ↔ ∀ 𝑣 ( ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) → 𝑣 ∈ 𝑧 ) ) ) |
28 |
27
|
rspccv |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → ( 𝑧 ∈ 𝑥 → ∀ 𝑣 ( ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) → 𝑣 ∈ 𝑧 ) ) ) |
29 |
|
psseq2 |
⊢ ( 𝑡 = 𝑤 → ( 𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑤 ) ) |
30 |
29
|
anbi1d |
⊢ ( 𝑡 = 𝑤 → ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) ↔ ( 𝑢 ⊊ 𝑤 ∧ Tr 𝑢 ) ) ) |
31 |
|
elequ2 |
⊢ ( 𝑡 = 𝑤 → ( 𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑤 ) ) |
32 |
30 31
|
imbi12d |
⊢ ( 𝑡 = 𝑤 → ( ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ↔ ( ( 𝑢 ⊊ 𝑤 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑤 ) ) ) |
33 |
32
|
albidv |
⊢ ( 𝑡 = 𝑤 → ( ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ↔ ∀ 𝑢 ( ( 𝑢 ⊊ 𝑤 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑤 ) ) ) |
34 |
|
psseq1 |
⊢ ( 𝑢 = 𝑦 → ( 𝑢 ⊊ 𝑤 ↔ 𝑦 ⊊ 𝑤 ) ) |
35 |
|
treq |
⊢ ( 𝑢 = 𝑦 → ( Tr 𝑢 ↔ Tr 𝑦 ) ) |
36 |
34 35
|
anbi12d |
⊢ ( 𝑢 = 𝑦 → ( ( 𝑢 ⊊ 𝑤 ∧ Tr 𝑢 ) ↔ ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) ) ) |
37 |
|
elequ1 |
⊢ ( 𝑢 = 𝑦 → ( 𝑢 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤 ) ) |
38 |
36 37
|
imbi12d |
⊢ ( 𝑢 = 𝑦 → ( ( ( 𝑢 ⊊ 𝑤 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑤 ) ↔ ( ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑤 ) ) ) |
39 |
38
|
cbvalvw |
⊢ ( ∀ 𝑢 ( ( 𝑢 ⊊ 𝑤 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑤 ) ↔ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑤 ) ) |
40 |
33 39
|
bitrdi |
⊢ ( 𝑡 = 𝑤 → ( ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) ↔ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑤 ) ) ) |
41 |
40
|
rspccv |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → ( 𝑤 ∈ 𝑥 → ∀ 𝑦 ( ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑤 ) ) ) |
42 |
28 41
|
anim12d |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( ∀ 𝑣 ( ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) → 𝑣 ∈ 𝑧 ) ∧ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑤 ) ) ) ) |
43 |
|
vex |
⊢ 𝑧 ∈ V |
44 |
|
vex |
⊢ 𝑤 ∈ V |
45 |
43 44
|
dfon2lem5 |
⊢ ( ( ∀ 𝑣 ( ( 𝑣 ⊊ 𝑧 ∧ Tr 𝑣 ) → 𝑣 ∈ 𝑧 ) ∧ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑤 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑤 ) ) → ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) |
46 |
42 45
|
syl6 |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → ( ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥 ) → ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) ) |
47 |
46
|
ralrimivv |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) |
48 |
15 47
|
jca |
⊢ ( ∀ 𝑡 ∈ 𝑥 ∀ 𝑢 ( ( 𝑢 ⊊ 𝑡 ∧ Tr 𝑢 ) → 𝑢 ∈ 𝑡 ) → ( E Fr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) ) |
49 |
14 48
|
syl |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → ( E Fr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) ) |
50 |
|
dfwe2 |
⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧 ) ) ) |
51 |
|
epel |
⊢ ( 𝑧 E 𝑤 ↔ 𝑧 ∈ 𝑤 ) |
52 |
|
biid |
⊢ ( 𝑧 = 𝑤 ↔ 𝑧 = 𝑤 ) |
53 |
|
epel |
⊢ ( 𝑤 E 𝑧 ↔ 𝑤 ∈ 𝑧 ) |
54 |
51 52 53
|
3orbi123i |
⊢ ( ( 𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧 ) ↔ ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) |
55 |
54
|
2ralbii |
⊢ ( ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧 ) ↔ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) |
56 |
55
|
anbi2i |
⊢ ( ( E Fr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧 ) ) ↔ ( E Fr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) ) |
57 |
50 56
|
bitri |
⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧 ) ) ) |
58 |
49 57
|
sylibr |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → E We 𝑥 ) |
59 |
|
df-ord |
⊢ ( Ord 𝑥 ↔ ( Tr 𝑥 ∧ E We 𝑥 ) ) |
60 |
12 58 59
|
sylanbrc |
⊢ ( ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) → Ord 𝑥 ) |
61 |
8 60
|
impbii |
⊢ ( Ord 𝑥 ↔ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) ) |
62 |
61
|
abbii |
⊢ { 𝑥 ∣ Ord 𝑥 } = { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) } |
63 |
1 62
|
eqtri |
⊢ On = { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦 ⊊ 𝑥 ∧ Tr 𝑦 ) → 𝑦 ∈ 𝑥 ) } |