| Step | Hyp | Ref
| Expression |
| 1 | | r1funlim 9806 |
. . . . 5
⊢ (Fun
𝑅1 ∧ Lim dom 𝑅1) |
| 2 | 1 | simpri 485 |
. . . 4
⊢ Lim dom
𝑅1 |
| 3 | | limord 6444 |
. . . 4
⊢ (Lim dom
𝑅1 → Ord dom 𝑅1) |
| 4 | 2, 3 | ax-mp 5 |
. . 3
⊢ Ord dom
𝑅1 |
| 5 | | ordelon 6408 |
. . 3
⊢ ((Ord dom
𝑅1 ∧ 𝐴 ∈ dom 𝑅1) →
𝐴 ∈
On) |
| 6 | 4, 5 | mpan 690 |
. 2
⊢ (𝐴 ∈ dom
𝑅1 → 𝐴 ∈ On) |
| 7 | | eleq1 2829 |
. . . 4
⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔
𝑦 ∈ dom
𝑅1)) |
| 8 | | eleq1 2829 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪
(𝑅1 “ On) ↔ 𝑦 ∈ ∪
(𝑅1 “ On))) |
| 9 | | fveq2 6906 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦)) |
| 10 | | id 22 |
. . . . . 6
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) |
| 11 | 9, 10 | eqeq12d 2753 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦)) |
| 12 | 8, 11 | anbi12d 632 |
. . . 4
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
| 13 | 7, 12 | imbi12d 344 |
. . 3
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)))) |
| 14 | | eleq1 2829 |
. . . 4
⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔
𝐴 ∈ dom
𝑅1)) |
| 15 | | eleq1 2829 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪
(𝑅1 “ On) ↔ 𝐴 ∈ ∪
(𝑅1 “ On))) |
| 16 | | fveq2 6906 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) |
| 17 | | id 22 |
. . . . . 6
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
| 18 | 16, 17 | eqeq12d 2753 |
. . . . 5
⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴)) |
| 19 | 15, 18 | anbi12d 632 |
. . . 4
⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))) |
| 20 | 14, 19 | imbi12d 344 |
. . 3
⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 →
(𝐴 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝐴) = 𝐴)))) |
| 21 | | ordtr1 6427 |
. . . . . . . . . 10
⊢ (Ord dom
𝑅1 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) →
𝑦 ∈ dom
𝑅1)) |
| 22 | 4, 21 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) →
𝑦 ∈ dom
𝑅1) |
| 23 | 22 | ancoms 458 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom
𝑅1) |
| 24 | | pm5.5 361 |
. . . . . . . 8
⊢ (𝑦 ∈ dom
𝑅1 → ((𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
| 25 | 23, 24 | syl 17 |
. . . . . . 7
⊢ ((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
| 26 | 25 | ralbidva 3176 |
. . . . . 6
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))) |
| 27 | | simplr 769 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥) |
| 28 | | ordelon 6408 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Ord dom
𝑅1 ∧ 𝑥 ∈ dom 𝑅1) →
𝑥 ∈
On) |
| 29 | 4, 28 | mpan 690 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ dom
𝑅1 → 𝑥 ∈ On) |
| 30 | 29 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On) |
| 31 | | eloni 6394 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ On → Ord 𝑥) |
| 32 | 30, 31 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥) |
| 33 | | ordelsuc 7840 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥)) |
| 34 | 27, 32, 33 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥)) |
| 35 | 27, 34 | mpbid 232 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ⊆ 𝑥) |
| 36 | 23 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom
𝑅1) |
| 37 | | limsuc 7870 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Lim dom
𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc
𝑦 ∈ dom
𝑅1)) |
| 38 | 2, 37 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ dom
𝑅1 ↔ suc 𝑦 ∈ dom
𝑅1) |
| 39 | 36, 38 | sylib 218 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom
𝑅1) |
| 40 | | simpll 767 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom
𝑅1) |
| 41 | | r1ord3g 9819 |
. . . . . . . . . . . . . . . . . 18
⊢ ((suc
𝑦 ∈ dom
𝑅1 ∧ 𝑥 ∈ dom 𝑅1) →
(suc 𝑦 ⊆ 𝑥 →
(𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥))) |
| 42 | 39, 40, 41 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc
𝑦) ⊆
(𝑅1‘𝑥))) |
| 43 | 35, 42 | mpd 15 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc
𝑦) ⊆
(𝑅1‘𝑥)) |
| 44 | | rankidb 9840 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈
(𝑅1‘suc (rank‘𝑦))) |
| 45 | 44 | ad2antrl 728 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc
(rank‘𝑦))) |
| 46 | | suceq 6450 |
. . . . . . . . . . . . . . . . . . 19
⊢
((rank‘𝑦) =
𝑦 → suc
(rank‘𝑦) = suc 𝑦) |
| 47 | 46 | ad2antll 729 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦) |
| 48 | 47 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc
(rank‘𝑦)) =
(𝑅1‘suc 𝑦)) |
| 49 | 45, 48 | eleqtrd 2843 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc
𝑦)) |
| 50 | 43, 49 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘𝑥)) |
| 51 | 50 | ex 412 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ dom
𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1‘𝑥))) |
| 52 | 51 | ralimdva 3167 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))) |
| 53 | 52 | imp 406 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)) |
| 54 | | dfss3 3972 |
. . . . . . . . . . . 12
⊢ (𝑥 ⊆
(𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)) |
| 55 | 53, 54 | sylibr 234 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1‘𝑥)) |
| 56 | | vex 3484 |
. . . . . . . . . . . 12
⊢ 𝑥 ∈ V |
| 57 | 56 | elpw 4604 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝒫
(𝑅1‘𝑥) ↔ 𝑥 ⊆ (𝑅1‘𝑥)) |
| 58 | 55, 57 | sylibr 234 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫
(𝑅1‘𝑥)) |
| 59 | | r1sucg 9809 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ dom
𝑅1 → (𝑅1‘suc 𝑥) = 𝒫
(𝑅1‘𝑥)) |
| 60 | 59 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc
𝑥) = 𝒫
(𝑅1‘𝑥)) |
| 61 | 58, 60 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc
𝑥)) |
| 62 | | r1elwf 9836 |
. . . . . . . . 9
⊢ (𝑥 ∈
(𝑅1‘suc 𝑥) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
| 63 | 61, 62 | syl 17 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
| 64 | | rankval3b 9866 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∪ (𝑅1 “ On) →
(rank‘𝑥) = ∩ {𝑧
∈ On ∣ ∀𝑦
∈ 𝑥 (rank‘𝑦) ∈ 𝑧}) |
| 65 | 63, 64 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧}) |
| 66 | | eleq1 2829 |
. . . . . . . . . . . . . . . 16
⊢
((rank‘𝑦) =
𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
| 67 | 66 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
| 68 | 67 | ralimi 3083 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
| 69 | | ralbi 3103 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)) |
| 70 | 68, 69 | syl 17 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)) |
| 71 | | dfss3 3972 |
. . . . . . . . . . . . 13
⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧) |
| 72 | 70, 71 | bitr4di 289 |
. . . . . . . . . . . 12
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧)) |
| 73 | 72 | rabbidv 3444 |
. . . . . . . . . . 11
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧}) |
| 74 | 73 | inteqd 4951 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧}) |
| 75 | 74 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧}) |
| 76 | 29 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On) |
| 77 | | intmin 4968 |
. . . . . . . . . 10
⊢ (𝑥 ∈ On → ∩ {𝑧
∈ On ∣ 𝑥 ⊆
𝑧} = 𝑥) |
| 78 | 76, 77 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥) |
| 79 | 65, 75, 78 | 3eqtrd 2781 |
. . . . . . . 8
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥) |
| 80 | 63, 79 | jca 511 |
. . . . . . 7
⊢ ((𝑥 ∈ dom
𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) |
| 81 | 80 | ex 412 |
. . . . . 6
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))) |
| 82 | 26, 81 | sylbid 240 |
. . . . 5
⊢ (𝑥 ∈ dom
𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))) |
| 83 | 82 | com12 32 |
. . . 4
⊢
(∀𝑦 ∈
𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥))) |
| 84 | 83 | a1i 11 |
. . 3
⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 →
(𝑦 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 →
(𝑥 ∈ ∪ (𝑅1 “ On) ∧
(rank‘𝑥) = 𝑥)))) |
| 85 | 13, 20, 84 | tfis3 7879 |
. 2
⊢ (𝐴 ∈ On → (𝐴 ∈ dom
𝑅1 → (𝐴 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))) |
| 86 | 6, 85 | mpcom 38 |
1
⊢ (𝐴 ∈ dom
𝑅1 → (𝐴 ∈ ∪
(𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)) |