Step | Hyp | Ref
| Expression |
1 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐵 ↔ 𝑦 ≼ 𝐵)) |
2 | | eleq1 2676 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈)) |
3 | 1, 2 | imbi12d 333 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈) ↔ (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈))) |
4 | 3 | imbi2d 329 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈)))) |
5 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑥 ≼ 𝐵 ↔ 𝐴 ≼ 𝐵)) |
6 | | eleq1 2676 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)) |
7 | 5, 6 | imbi12d 333 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → ((𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈) ↔ (𝐴 ≼ 𝐵 → 𝐴 ∈ 𝑈))) |
8 | 7 | imbi2d 329 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝐴 ≼ 𝐵 → 𝐴 ∈ 𝑈)))) |
9 | | r19.21v 2943 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝑥 ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → ∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈))) |
10 | | simpl1 1057 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → 𝑥 ∈ On) |
11 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑥 ∈ V |
12 | | onelss 5683 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥)) |
13 | 12 | imp 444 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥) |
14 | | ssdomg 7887 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ V → (𝑦 ⊆ 𝑥 → 𝑦 ≼ 𝑥)) |
15 | 11, 13, 14 | mpsyl 66 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥) |
16 | 10, 15 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥) |
17 | | simplr 788 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ≼ 𝐵) |
18 | | domtr 7895 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ≼ 𝑥 ∧ 𝑥 ≼ 𝐵) → 𝑦 ≼ 𝐵) |
19 | 16, 17, 18 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝐵) |
20 | | pm2.27 41 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ≼ 𝐵 → ((𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈)) |
21 | 19, 20 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) ∧ 𝑦 ∈ 𝑥) → ((𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈)) |
22 | 21 | ralimdva 2945 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑈)) |
23 | | dfss3 3558 |
. . . . . . . . . . . . 13
⊢ (𝑥 ⊆ 𝑈 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑈) |
24 | | domeng 7855 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ 𝑈 → (𝑥 ≼ 𝐵 ↔ ∃𝑦(𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵))) |
25 | 24 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑥 ≼ 𝐵 ↔ ∃𝑦(𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵))) |
26 | 25 | biimpa 500 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → ∃𝑦(𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵)) |
27 | | simpl2 1058 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → 𝑈 ∈ Univ) |
28 | | gruss 9497 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑦 ⊆ 𝐵) → 𝑦 ∈ 𝑈) |
29 | 28 | 3expia 1259 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦 ⊆ 𝐵 → 𝑦 ∈ 𝑈)) |
30 | 29 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦 ⊆ 𝐵 → 𝑦 ∈ 𝑈)) |
31 | 30 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (𝑦 ⊆ 𝐵 → 𝑦 ∈ 𝑈)) |
32 | | ensym 7891 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥) |
33 | 32 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥)) |
34 | 31, 33 | anim12d 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → ((𝑦 ⊆ 𝐵 ∧ 𝑥 ≈ 𝑦) → (𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥))) |
35 | 34 | ancomsd 469 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → ((𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵) → (𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥))) |
36 | 35 | eximdv 1833 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (∃𝑦(𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵) → ∃𝑦(𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥))) |
37 | | gruen 9513 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑈 ∈ Univ ∧ 𝑥 ⊆ 𝑈 ∧ (𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥)) → 𝑥 ∈ 𝑈) |
38 | 37 | 3com23 1263 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈 ∈ Univ ∧ (𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥) ∧ 𝑥 ⊆ 𝑈) → 𝑥 ∈ 𝑈) |
39 | 38 | 3exp 1256 |
. . . . . . . . . . . . . . . 16
⊢ (𝑈 ∈ Univ → ((𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥) → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈))) |
40 | 39 | exlimdv 1848 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ Univ →
(∃𝑦(𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥) → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈))) |
41 | 27, 36, 40 | sylsyld 59 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (∃𝑦(𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈))) |
42 | 26, 41 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) |
43 | 23, 42 | syl5bir 232 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑈 → 𝑥 ∈ 𝑈)) |
44 | 22, 43 | syld 46 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) ∧ 𝑥 ≼ 𝐵) → (∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈)) |
45 | 44 | ex 449 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑥 ≼ 𝐵 → (∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))) |
46 | 45 | com23 84 |
. . . . . . . . 9
⊢ ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → (𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈))) |
47 | 46 | 3expib 1260 |
. . . . . . . 8
⊢ (𝑥 ∈ On → ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈) → (𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈)))) |
48 | 47 | a2d 29 |
. . . . . . 7
⊢ (𝑥 ∈ On → (((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → ∀𝑦 ∈ 𝑥 (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈)) → ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈)))) |
49 | 9, 48 | syl5bi 231 |
. . . . . 6
⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈)) → ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈)))) |
50 | 4, 8, 49 | tfis3 6949 |
. . . . 5
⊢ (𝐴 ∈ On → ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝐴 ≼ 𝐵 → 𝐴 ∈ 𝑈))) |
51 | 50 | com3l 87 |
. . . 4
⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝐴 ≼ 𝐵 → (𝐴 ∈ On → 𝐴 ∈ 𝑈))) |
52 | 51 | impr 647 |
. . 3
⊢ ((𝑈 ∈ Univ ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵)) → (𝐴 ∈ On → 𝐴 ∈ 𝑈)) |
53 | 52 | 3impia 1253 |
. 2
⊢ ((𝑈 ∈ Univ ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵) ∧ 𝐴 ∈ On) → 𝐴 ∈ 𝑈) |
54 | 53 | 3com23 1263 |
1
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵)) → 𝐴 ∈ 𝑈) |