Step | Hyp | Ref
| Expression |
1 | | pssss 3664 |
. . . 4
⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴) |
2 | | ssexg 4732 |
. . . 4
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ V) |
3 | 1, 2 | sylan 487 |
. . 3
⊢ ((𝐵 ⊊ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ V) |
4 | 3 | ancoms 468 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ∈ V) |
5 | | psseq2 3657 |
. . . . . . . 8
⊢ (𝑧 = ∅ → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ ∅)) |
6 | | rexeq 3116 |
. . . . . . . 8
⊢ (𝑧 = ∅ → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)) |
7 | 5, 6 | imbi12d 333 |
. . . . . . 7
⊢ (𝑧 = ∅ → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥))) |
8 | 7 | albidv 1836 |
. . . . . 6
⊢ (𝑧 = ∅ → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥))) |
9 | | psseq2 3657 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝑦)) |
10 | | rexeq 3116 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) |
11 | 9, 10 | imbi12d 333 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥))) |
12 | 11 | albidv 1836 |
. . . . . 6
⊢ (𝑧 = 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥))) |
13 | | psseq2 3657 |
. . . . . . . 8
⊢ (𝑧 = suc 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ suc 𝑦)) |
14 | | rexeq 3116 |
. . . . . . . 8
⊢ (𝑧 = suc 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)) |
15 | 13, 14 | imbi12d 333 |
. . . . . . 7
⊢ (𝑧 = suc 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))) |
16 | 15 | albidv 1836 |
. . . . . 6
⊢ (𝑧 = suc 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))) |
17 | | psseq2 3657 |
. . . . . . . 8
⊢ (𝑧 = 𝐴 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝐴)) |
18 | | rexeq 3116 |
. . . . . . . 8
⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)) |
19 | 17, 18 | imbi12d 333 |
. . . . . . 7
⊢ (𝑧 = 𝐴 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))) |
20 | 19 | albidv 1836 |
. . . . . 6
⊢ (𝑧 = 𝐴 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))) |
21 | | npss0 3966 |
. . . . . . . 8
⊢ ¬
𝑤 ⊊
∅ |
22 | 21 | pm2.21i 115 |
. . . . . . 7
⊢ (𝑤 ⊊ ∅ →
∃𝑥 ∈ ∅
𝑤 ≈ 𝑥) |
23 | 22 | ax-gen 1713 |
. . . . . 6
⊢
∀𝑤(𝑤 ⊊ ∅ →
∃𝑥 ∈ ∅
𝑤 ≈ 𝑥) |
24 | | nfv 1830 |
. . . . . . 7
⊢
Ⅎ𝑤 𝑦 ∈ ω |
25 | | nfa1 2015 |
. . . . . . 7
⊢
Ⅎ𝑤∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) |
26 | | elequ1 1984 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤)) |
27 | 26 | biimpcd 238 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑦 ∈ 𝑤)) |
28 | 27 | con3d 147 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ 𝑤 → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦)) |
29 | 28 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦)) |
30 | | pssss 3664 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 ⊊ suc 𝑦 → 𝑤 ⊆ suc 𝑦) |
31 | 30 | sseld 3567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 ⊊ suc 𝑦 → (𝑧 ∈ 𝑤 → 𝑧 ∈ suc 𝑦)) |
32 | | elsuci 5708 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 ∈ suc 𝑦 → (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦)) |
33 | 32 | ord 391 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 ∈ suc 𝑦 → (¬ 𝑧 ∈ 𝑦 → 𝑧 = 𝑦)) |
34 | 33 | con1d 138 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 ∈ suc 𝑦 → (¬ 𝑧 = 𝑦 → 𝑧 ∈ 𝑦)) |
35 | 31, 34 | syl6 34 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ⊊ suc 𝑦 → (𝑧 ∈ 𝑤 → (¬ 𝑧 = 𝑦 → 𝑧 ∈ 𝑦))) |
36 | 35 | imp 444 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑧 = 𝑦 → 𝑧 ∈ 𝑦)) |
37 | 29, 36 | syld 46 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑦 ∈ 𝑤 → 𝑧 ∈ 𝑦)) |
38 | 37 | impancom 455 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦)) |
39 | 38 | ssrdv 3574 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → 𝑤 ⊆ 𝑦) |
40 | 39 | anim1i 590 |
. . . . . . . . . . . . . . 15
⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦)) |
41 | | dfpss2 3654 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ⊊ 𝑦 ↔ (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦)) |
42 | 40, 41 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → 𝑤 ⊊ 𝑦) |
43 | | elelsuc 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ suc 𝑦) |
44 | 43 | anim1i 590 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ 𝑦 ∧ 𝑤 ≈ 𝑥) → (𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑥)) |
45 | 44 | reximi2 2993 |
. . . . . . . . . . . . . 14
⊢
(∃𝑥 ∈
𝑦 𝑤 ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥) |
46 | 42, 45 | imim12i 60 |
. . . . . . . . . . . . 13
⊢ ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)) |
47 | 46 | exp4c 634 |
. . . . . . . . . . . 12
⊢ ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))) |
48 | 47 | sps 2043 |
. . . . . . . . . . 11
⊢
(∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))) |
49 | 48 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ω ∧
∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))) |
50 | 49 | com4t 91 |
. . . . . . . . 9
⊢ (¬
𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))) |
51 | | anidm 674 |
. . . . . . . . . . . . . 14
⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) ↔ 𝑤 ⊊ suc 𝑦) |
52 | | ssdif 3707 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ⊆ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦})) |
53 | | nnord 6965 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ω → Ord 𝑦) |
54 | | orddif 5737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (Ord
𝑦 → 𝑦 = (suc 𝑦 ∖ {𝑦})) |
55 | 53, 54 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ω → 𝑦 = (suc 𝑦 ∖ {𝑦})) |
56 | 55 | sseq2d 3596 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ω → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦}))) |
57 | 52, 56 | syl5ibr 235 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ω → (𝑤 ⊆ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ 𝑦)) |
58 | 30, 57 | syl5 33 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ω → (𝑤 ⊊ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ 𝑦)) |
59 | | pssnel 3991 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ⊊ suc 𝑦 → ∃𝑧(𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤)) |
60 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ (𝑤 ∖ {𝑦}) ↔ 𝑧 ∈ 𝑦)) |
61 | | eldifi 3694 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 ∈ (𝑤 ∖ {𝑦}) → 𝑧 ∈ 𝑤) |
62 | 60, 61 | syl6bir 243 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)) |
63 | 62 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)) |
64 | | eleq1a 2683 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑧 ∈ 𝑤)) |
65 | 33, 64 | sylan9r 688 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → (¬ 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)) |
66 | 65 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → (¬ 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)) |
67 | 63, 66 | pm2.61d 169 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → 𝑧 ∈ 𝑤) |
68 | 67 | ex 449 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → ((𝑤 ∖ {𝑦}) = 𝑦 → 𝑧 ∈ 𝑤)) |
69 | 68 | con3d 147 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → (¬ 𝑧 ∈ 𝑤 → ¬ (𝑤 ∖ {𝑦}) = 𝑦)) |
70 | 69 | expimpd 627 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ 𝑤 → ((𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤) → ¬ (𝑤 ∖ {𝑦}) = 𝑦)) |
71 | 70 | exlimdv 1848 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝑤 → (∃𝑧(𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤) → ¬ (𝑤 ∖ {𝑦}) = 𝑦)) |
72 | 59, 71 | syl5 33 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝑤 → (𝑤 ⊊ suc 𝑦 → ¬ (𝑤 ∖ {𝑦}) = 𝑦)) |
73 | 58, 72 | im2anan9r 877 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦))) |
74 | 51, 73 | syl5bir 232 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦))) |
75 | | dfpss2 3654 |
. . . . . . . . . . . . 13
⊢ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 ↔ ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)) |
76 | 74, 75 | syl6ibr 241 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊊ 𝑦)) |
77 | | psseq1 3656 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑧 → (𝑤 ⊊ 𝑦 ↔ 𝑧 ⊊ 𝑦)) |
78 | | breq1 4586 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑧 ≈ 𝑥)) |
79 | 78 | rexbidv 3034 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥)) |
80 | 77, 79 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑧 → ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ (𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))) |
81 | 80 | cbvalv 2261 |
. . . . . . . . . . . . 13
⊢
(∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ ∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥)) |
82 | | vex 3176 |
. . . . . . . . . . . . . . 15
⊢ 𝑤 ∈ V |
83 | | difss 3699 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∖ {𝑦}) ⊆ 𝑤 |
84 | 82, 83 | ssexi 4731 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∖ {𝑦}) ∈ V |
85 | | psseq1 3656 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ⊊ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊊ 𝑦)) |
86 | | breq1 4586 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ≈ 𝑥 ↔ (𝑤 ∖ {𝑦}) ≈ 𝑥)) |
87 | 86 | rexbidv 3034 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)) |
88 | 85, 87 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑤 ∖ {𝑦}) → ((𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) ↔ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))) |
89 | 84, 88 | spcv 3272 |
. . . . . . . . . . . . 13
⊢
(∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)) |
90 | 81, 89 | sylbi 206 |
. . . . . . . . . . . 12
⊢
(∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)) |
91 | 76, 90 | sylan9 687 |
. . . . . . . . . . 11
⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)) |
92 | | ordsucelsuc 6914 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Ord
𝑦 → (𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ suc 𝑦)) |
93 | 92 | biimpd 218 |
. . . . . . . . . . . . . . . . . . 19
⊢ (Ord
𝑦 → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)) |
94 | 53, 93 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)) |
95 | 94 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)) |
96 | 95 | adantrd 483 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → suc 𝑥 ∈ suc 𝑦)) |
97 | | elnn 6967 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) |
98 | | snex 4835 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
{〈𝑦, 𝑥〉} ∈
V |
99 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 𝑦 ∈ V |
100 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 𝑥 ∈ V |
101 | 99, 100 | f1osn 6088 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
{〈𝑦, 𝑥〉}:{𝑦}–1-1-onto→{𝑥} |
102 | | f1oen3g 7857 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(({〈𝑦, 𝑥〉} ∈ V ∧
{〈𝑦, 𝑥〉}:{𝑦}–1-1-onto→{𝑥}) → {𝑦} ≈ {𝑥}) |
103 | 98, 101, 102 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑦} ≈ {𝑥} |
104 | 103 | jctr 563 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑤 ∖ {𝑦}) ≈ 𝑥 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥})) |
105 | | nnord 6965 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ ω → Ord 𝑥) |
106 | | orddisj 5679 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (Ord
𝑥 → (𝑥 ∩ {𝑥}) = ∅) |
107 | 105, 106 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅) |
108 | | incom 3767 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({𝑦} ∩ (𝑤 ∖ {𝑦})) = ((𝑤 ∖ {𝑦}) ∩ {𝑦}) |
109 | | disjdif 3992 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({𝑦} ∩ (𝑤 ∖ {𝑦})) = ∅ |
110 | 108, 109 | eqtr3i 2634 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ |
111 | 107, 110 | jctil 558 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ω → (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) |
112 | | unen 7925 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}) ∧ (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥})) |
113 | 104, 111,
112 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥})) |
114 | | difsnid 4282 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) = 𝑤) |
115 | 114 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ 𝑤 → 𝑤 = ((𝑤 ∖ {𝑦}) ∪ {𝑦})) |
116 | | df-suc 5646 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ suc 𝑥 = (𝑥 ∪ {𝑥}) |
117 | 116 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ 𝑤 → suc 𝑥 = (𝑥 ∪ {𝑥})) |
118 | 115, 117 | breq12d 4596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ 𝑤 → (𝑤 ≈ suc 𝑥 ↔ ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))) |
119 | 113, 118 | syl5ibr 235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑤 ≈ suc 𝑥)) |
120 | 97, 119 | sylan2i 685 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω)) → 𝑤 ≈ suc 𝑥)) |
121 | 120 | exp4d 635 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 → (𝑥 ∈ 𝑦 → (𝑦 ∈ ω → 𝑤 ≈ suc 𝑥)))) |
122 | 121 | com24 93 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ 𝑤 → (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 → 𝑤 ≈ suc 𝑥)))) |
123 | 122 | imp4b 611 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → 𝑤 ≈ suc 𝑥)) |
124 | 96, 123 | jcad 554 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → (suc 𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥))) |
125 | | breq2 4587 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = suc 𝑥 → (𝑤 ≈ 𝑧 ↔ 𝑤 ≈ suc 𝑥)) |
126 | 125 | rspcev 3282 |
. . . . . . . . . . . . . . 15
⊢ ((suc
𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧) |
127 | 124, 126 | syl6 34 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)) |
128 | 127 | exlimdv 1848 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)) |
129 | | df-rex 2902 |
. . . . . . . . . . . . 13
⊢
(∃𝑥 ∈
𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥)) |
130 | | breq2 4587 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑧)) |
131 | 130 | cbvrexv 3148 |
. . . . . . . . . . . . 13
⊢
(∃𝑥 ∈ suc
𝑦𝑤 ≈ 𝑥 ↔ ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧) |
132 | 128, 129,
131 | 3imtr4g 284 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)) |
133 | 132 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)) |
134 | 91, 133 | syld 46 |
. . . . . . . . . 10
⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)) |
135 | 134 | expl 646 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝑤 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))) |
136 | 82 | eqelsuc 5723 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑦 → 𝑤 ∈ suc 𝑦) |
137 | 82 | enref 7874 |
. . . . . . . . . . 11
⊢ 𝑤 ≈ 𝑤 |
138 | | breq2 4587 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑤)) |
139 | 138 | rspcev 3282 |
. . . . . . . . . . 11
⊢ ((𝑤 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑤) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥) |
140 | 136, 137,
139 | sylancl 693 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥) |
141 | 140 | 2a1d 26 |
. . . . . . . . 9
⊢ (𝑤 = 𝑦 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))) |
142 | 50, 135, 141 | pm2.61ii 176 |
. . . . . . . 8
⊢ ((𝑦 ∈ ω ∧
∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)) |
143 | 142 | ex 449 |
. . . . . . 7
⊢ (𝑦 ∈ ω →
(∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))) |
144 | 24, 25, 143 | alrimd 2071 |
. . . . . 6
⊢ (𝑦 ∈ ω →
(∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))) |
145 | 8, 12, 16, 20, 23, 144 | finds 6984 |
. . . . 5
⊢ (𝐴 ∈ ω →
∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)) |
146 | | psseq1 3656 |
. . . . . . 7
⊢ (𝑤 = 𝐵 → (𝑤 ⊊ 𝐴 ↔ 𝐵 ⊊ 𝐴)) |
147 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑤 = 𝐵 → (𝑤 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥)) |
148 | 147 | rexbidv 3034 |
. . . . . . 7
⊢ (𝑤 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)) |
149 | 146, 148 | imbi12d 333 |
. . . . . 6
⊢ (𝑤 = 𝐵 → ((𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) ↔ (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))) |
150 | 149 | spcgv 3266 |
. . . . 5
⊢ (𝐵 ∈ V → (∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) → (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))) |
151 | 145, 150 | syl5 33 |
. . . 4
⊢ (𝐵 ∈ V → (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))) |
152 | 151 | com3l 87 |
. . 3
⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → (𝐵 ∈ V → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))) |
153 | 152 | imp 444 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (𝐵 ∈ V → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)) |
154 | 4, 153 | mpd 15 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥) |