Step | Hyp | Ref
| Expression |
1 | | unieq 4830 |
. . . . 5
⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪
𝐴) |
2 | 1 | fveq2d 6721 |
. . . 4
⊢ (𝑥 = 𝐴 → (rank‘∪ 𝑥) =
(rank‘∪ 𝐴)) |
3 | | fveq2 6717 |
. . . . 5
⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) |
4 | 3 | unieqd 4833 |
. . . 4
⊢ (𝑥 = 𝐴 → ∪
(rank‘𝑥) = ∪ (rank‘𝐴)) |
5 | 2, 4 | eqeq12d 2753 |
. . 3
⊢ (𝑥 = 𝐴 → ((rank‘∪ 𝑥) =
∪ (rank‘𝑥) ↔ (rank‘∪ 𝐴) =
∪ (rank‘𝐴))) |
6 | | vex 3412 |
. . . . . . 7
⊢ 𝑥 ∈ V |
7 | 6 | rankuni2 9471 |
. . . . . 6
⊢
(rank‘∪ 𝑥) = ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) |
8 | | fvex 6730 |
. . . . . . 7
⊢
(rank‘𝑧)
∈ V |
9 | 8 | dfiun2 4942 |
. . . . . 6
⊢ ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} |
10 | 7, 9 | eqtri 2765 |
. . . . 5
⊢
(rank‘∪ 𝑥) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} |
11 | | df-rex 3067 |
. . . . . . . 8
⊢
(∃𝑧 ∈
𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧))) |
12 | 6 | rankel 9455 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝑥 → (rank‘𝑧) ∈ (rank‘𝑥)) |
13 | 12 | anim1i 618 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧))) |
14 | 13 | eximi 1842 |
. . . . . . . . 9
⊢
(∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧))) |
15 | | 19.42v 1962 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧))) |
16 | | eleq1 2825 |
. . . . . . . . . . . 12
⊢ (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥))) |
17 | 16 | pm5.32ri 579 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧))) |
18 | 17 | exbii 1855 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧))) |
19 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥)) |
20 | | rankon 9411 |
. . . . . . . . . . . . . . . . 17
⊢
(rank‘𝑥)
∈ On |
21 | 20 | oneli 6321 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On) |
22 | | r1fnon 9383 |
. . . . . . . . . . . . . . . . 17
⊢
𝑅1 Fn On |
23 | | fndm 6481 |
. . . . . . . . . . . . . . . . 17
⊢
(𝑅1 Fn On → dom 𝑅1 =
On) |
24 | 22, 23 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ dom
𝑅1 = On |
25 | 21, 24 | eleqtrrdi 2849 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom
𝑅1) |
26 | | rankr1id 9478 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ dom
𝑅1 ↔ (rank‘(𝑅1‘𝑦)) = 𝑦) |
27 | 25, 26 | sylib 221 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (rank‘𝑥) →
(rank‘(𝑅1‘𝑦)) = 𝑦) |
28 | 27 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 =
(rank‘(𝑅1‘𝑦))) |
29 | | fvex 6730 |
. . . . . . . . . . . . . 14
⊢
(𝑅1‘𝑦) ∈ V |
30 | | fveq2 6717 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 =
(𝑅1‘𝑦) → (rank‘𝑧) =
(rank‘(𝑅1‘𝑦))) |
31 | 30 | eqeq2d 2748 |
. . . . . . . . . . . . . 14
⊢ (𝑧 =
(𝑅1‘𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 =
(rank‘(𝑅1‘𝑦)))) |
32 | 29, 31 | spcev 3521 |
. . . . . . . . . . . . 13
⊢ (𝑦 =
(rank‘(𝑅1‘𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧)) |
33 | 28, 32 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧)) |
34 | 33 | ancli 552 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (rank‘𝑥) → (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧))) |
35 | 19, 34 | impbii 212 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥)) |
36 | 15, 18, 35 | 3bitr3i 304 |
. . . . . . . . 9
⊢
(∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥)) |
37 | 14, 36 | sylib 221 |
. . . . . . . 8
⊢
(∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥)) |
38 | 11, 37 | sylbi 220 |
. . . . . . 7
⊢
(∃𝑧 ∈
𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥)) |
39 | 38 | abssi 3983 |
. . . . . 6
⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥) |
40 | 39 | unissi 4828 |
. . . . 5
⊢ ∪ {𝑦
∣ ∃𝑧 ∈
𝑥 𝑦 = (rank‘𝑧)} ⊆ ∪
(rank‘𝑥) |
41 | 10, 40 | eqsstri 3935 |
. . . 4
⊢
(rank‘∪ 𝑥) ⊆ ∪
(rank‘𝑥) |
42 | | pwuni 4858 |
. . . . . . . 8
⊢ 𝑥 ⊆ 𝒫 ∪ 𝑥 |
43 | | vuniex 7527 |
. . . . . . . . . 10
⊢ ∪ 𝑥
∈ V |
44 | 43 | pwex 5273 |
. . . . . . . . 9
⊢ 𝒫
∪ 𝑥 ∈ V |
45 | 44 | rankss 9465 |
. . . . . . . 8
⊢ (𝑥 ⊆ 𝒫 ∪ 𝑥
→ (rank‘𝑥)
⊆ (rank‘𝒫 ∪ 𝑥)) |
46 | 42, 45 | ax-mp 5 |
. . . . . . 7
⊢
(rank‘𝑥)
⊆ (rank‘𝒫 ∪ 𝑥) |
47 | 43 | rankpw 9459 |
. . . . . . 7
⊢
(rank‘𝒫 ∪ 𝑥) = suc (rank‘∪ 𝑥) |
48 | 46, 47 | sseqtri 3937 |
. . . . . 6
⊢
(rank‘𝑥)
⊆ suc (rank‘∪ 𝑥) |
49 | 48 | unissi 4828 |
. . . . 5
⊢ ∪ (rank‘𝑥) ⊆ ∪ suc
(rank‘∪ 𝑥) |
50 | | rankon 9411 |
. . . . . 6
⊢
(rank‘∪ 𝑥) ∈ On |
51 | 50 | onunisuci 6327 |
. . . . 5
⊢ ∪ suc (rank‘∪ 𝑥) = (rank‘∪ 𝑥) |
52 | 49, 51 | sseqtri 3937 |
. . . 4
⊢ ∪ (rank‘𝑥) ⊆ (rank‘∪ 𝑥) |
53 | 41, 52 | eqssi 3917 |
. . 3
⊢
(rank‘∪ 𝑥) = ∪
(rank‘𝑥) |
54 | 5, 53 | vtoclg 3481 |
. 2
⊢ (𝐴 ∈ V →
(rank‘∪ 𝐴) = ∪
(rank‘𝐴)) |
55 | | uniexb 7549 |
. . . . 5
⊢ (𝐴 ∈ V ↔ ∪ 𝐴
∈ V) |
56 | | fvprc 6709 |
. . . . 5
⊢ (¬
∪ 𝐴 ∈ V → (rank‘∪ 𝐴) =
∅) |
57 | 55, 56 | sylnbi 333 |
. . . 4
⊢ (¬
𝐴 ∈ V →
(rank‘∪ 𝐴) = ∅) |
58 | | uni0 4849 |
. . . 4
⊢ ∪ ∅ = ∅ |
59 | 57, 58 | eqtr4di 2796 |
. . 3
⊢ (¬
𝐴 ∈ V →
(rank‘∪ 𝐴) = ∪
∅) |
60 | | fvprc 6709 |
. . . 4
⊢ (¬
𝐴 ∈ V →
(rank‘𝐴) =
∅) |
61 | 60 | unieqd 4833 |
. . 3
⊢ (¬
𝐴 ∈ V → ∪ (rank‘𝐴) = ∪
∅) |
62 | 59, 61 | eqtr4d 2780 |
. 2
⊢ (¬
𝐴 ∈ V →
(rank‘∪ 𝐴) = ∪
(rank‘𝐴)) |
63 | 54, 62 | pm2.61i 185 |
1
⊢
(rank‘∪ 𝐴) = ∪
(rank‘𝐴) |