| Step | Hyp | Ref
| Expression |
| 1 | | simpl 482 |
. . . 4
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ 𝐴 ∈
ω) |
| 2 | | nnon 7893 |
. . . 4
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ 𝐴 ∈
On) |
| 4 | | omelon 9686 |
. . . . 5
⊢ ω
∈ On |
| 5 | | limom 7903 |
. . . . 5
⊢ Lim
ω |
| 6 | 4, 5 | pm3.2i 470 |
. . . 4
⊢ (ω
∈ On ∧ Lim ω) |
| 7 | 6 | a1i 11 |
. . 3
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (ω ∈ On ∧ Lim ω)) |
| 8 | | 0elon 6438 |
. . . . 5
⊢ ∅
∈ On |
| 9 | 8 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ ∅ ∈ On) |
| 10 | | 0ss 4400 |
. . . . 5
⊢ ∅
⊆ 1o |
| 11 | 10 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ ∅ ⊆ 1o) |
| 12 | | simpr 484 |
. . . 4
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ 1o ∈ 𝐴) |
| 13 | | ontr2 6431 |
. . . . 5
⊢ ((∅
∈ On ∧ 𝐴 ∈
On) → ((∅ ⊆ 1o ∧ 1o ∈ 𝐴) → ∅ ∈ 𝐴)) |
| 14 | 13 | imp 406 |
. . . 4
⊢
(((∅ ∈ On ∧ 𝐴 ∈ On) ∧ (∅ ⊆
1o ∧ 1o ∈ 𝐴)) → ∅ ∈ 𝐴) |
| 15 | 9, 3, 11, 12, 14 | syl22anc 839 |
. . 3
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ ∅ ∈ 𝐴) |
| 16 | | oelim 8572 |
. . 3
⊢ (((𝐴 ∈ On ∧ (ω ∈
On ∧ Lim ω)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ω) = ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥)) |
| 17 | 3, 7, 15, 16 | syl21anc 838 |
. 2
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (𝐴
↑o ω) = ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥)) |
| 18 | | ovex 7464 |
. . . 4
⊢ (𝐴 ↑o 𝑥) ∈ V |
| 19 | 18 | dfiun2 5033 |
. . 3
⊢ ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} |
| 20 | | eluniab 4921 |
. . . . . 6
⊢ (𝑧 ∈ ∪ {𝑦
∣ ∃𝑥 ∈
ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥))) |
| 21 | | 19.42v 1953 |
. . . . . . . 8
⊢
(∃𝑥(𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) ↔ (𝑧 ∈ 𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))) |
| 22 | | 3anass 1095 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))) |
| 23 | 22 | exbii 1848 |
. . . . . . . 8
⊢
(∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥(𝑧 ∈ 𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))) |
| 24 | | df-rex 3071 |
. . . . . . . . 9
⊢
(∃𝑥 ∈
ω 𝑦 = (𝐴 ↑o 𝑥) ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) |
| 25 | 24 | anbi2i 623 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))) |
| 26 | 21, 23, 25 | 3bitr4ri 304 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) |
| 27 | 26 | exbii 1848 |
. . . . . 6
⊢
(∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑦∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) |
| 28 | | excom 2162 |
. . . . . 6
⊢
(∃𝑦∃𝑥(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) |
| 29 | 20, 27, 28 | 3bitri 297 |
. . . . 5
⊢ (𝑧 ∈ ∪ {𝑦
∣ ∃𝑥 ∈
ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) |
| 30 | | simpr3 1197 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑦 = (𝐴 ↑o 𝑥)) |
| 31 | | simp2 1138 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑥 ∈ ω) |
| 32 | | nnecl 8651 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ↑o 𝑥) ∈
ω) |
| 33 | 1, 31, 32 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → (𝐴 ↑o 𝑥) ∈ ω) |
| 34 | | onelss 6426 |
. . . . . . . . . . 11
⊢ (ω
∈ On → ((𝐴
↑o 𝑥)
∈ ω → (𝐴
↑o 𝑥)
⊆ ω)) |
| 35 | 4, 33, 34 | mpsyl 68 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → (𝐴 ↑o 𝑥) ⊆ ω) |
| 36 | 30, 35 | eqsstrd 4018 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑦 ⊆ ω) |
| 37 | | simpr1 1195 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑧 ∈ 𝑦) |
| 38 | 36, 37 | sseldd 3984 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ (𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) → 𝑧 ∈ ω) |
| 39 | 38 | ex 412 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑧 ∈ ω)) |
| 40 | 39 | exlimdvv 1934 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) → 𝑧 ∈ ω)) |
| 41 | | peano2 7912 |
. . . . . . . . 9
⊢ (𝑧 ∈ ω → suc 𝑧 ∈
ω) |
| 42 | 41 | adantl 481 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ suc 𝑧 ∈
ω) |
| 43 | | ovex 7464 |
. . . . . . . . . 10
⊢ (𝐴 ↑o suc 𝑧) ∈ V |
| 44 | 43 | a1i 11 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ (𝐴
↑o suc 𝑧)
∈ V) |
| 45 | 2 | anim1i 615 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (𝐴 ∈ On ∧
1o ∈ 𝐴)) |
| 46 | | ondif2 8540 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ (On ∖
2o) ↔ (𝐴
∈ On ∧ 1o ∈ 𝐴)) |
| 47 | 45, 46 | sylibr 234 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ 𝐴 ∈ (On ∖
2o)) |
| 48 | | nnon 7893 |
. . . . . . . . . . . . 13
⊢ (suc
𝑧 ∈ ω → suc
𝑧 ∈
On) |
| 49 | 41, 48 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ω → suc 𝑧 ∈ On) |
| 50 | | oeworde 8631 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (On ∖
2o) ∧ suc 𝑧
∈ On) → suc 𝑧
⊆ (𝐴
↑o suc 𝑧)) |
| 51 | 47, 49, 50 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ suc 𝑧 ⊆ (𝐴 ↑o suc 𝑧)) |
| 52 | | vex 3484 |
. . . . . . . . . . . . 13
⊢ 𝑧 ∈ V |
| 53 | 52 | sucid 6466 |
. . . . . . . . . . . 12
⊢ 𝑧 ∈ suc 𝑧 |
| 54 | 53 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ 𝑧 ∈ suc 𝑧) |
| 55 | 51, 54 | sseldd 3984 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ 𝑧 ∈ (𝐴 ↑o suc 𝑧)) |
| 56 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ (𝐴
↑o suc 𝑧) =
(𝐴 ↑o suc
𝑧)) |
| 57 | 55, 42, 56 | 3jca 1129 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ (𝑧 ∈ (𝐴 ↑o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧))) |
| 58 | | eleq2 2830 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ (𝐴 ↑o suc 𝑧))) |
| 59 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → (𝑦 = (𝐴 ↑o suc 𝑧) ↔ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧))) |
| 60 | 58, 59 | 3anbi13d 1440 |
. . . . . . . . 9
⊢ (𝑦 = (𝐴 ↑o suc 𝑧) → ((𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧)) ↔ (𝑧 ∈ (𝐴 ↑o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴 ↑o suc 𝑧) = (𝐴 ↑o suc 𝑧)))) |
| 61 | 44, 57, 60 | spcedv 3598 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ ∃𝑦(𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧))) |
| 62 | | eleq1 2829 |
. . . . . . . . . 10
⊢ (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω)) |
| 63 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑧 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑧)) |
| 64 | 63 | eqeq2d 2748 |
. . . . . . . . . 10
⊢ (𝑥 = suc 𝑧 → (𝑦 = (𝐴 ↑o 𝑥) ↔ 𝑦 = (𝐴 ↑o suc 𝑧))) |
| 65 | 62, 64 | 3anbi23d 1441 |
. . . . . . . . 9
⊢ (𝑥 = suc 𝑧 → ((𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧)))) |
| 66 | 65 | exbidv 1921 |
. . . . . . . 8
⊢ (𝑥 = suc 𝑧 → (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴 ↑o suc 𝑧)))) |
| 67 | 42, 61, 66 | spcedv 3598 |
. . . . . . 7
⊢ (((𝐴 ∈ ω ∧
1o ∈ 𝐴)
∧ 𝑧 ∈ ω)
→ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥))) |
| 68 | 67 | ex 412 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (𝑧 ∈ ω
→ ∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)))) |
| 69 | 40, 68 | impbid 212 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (∃𝑥∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑥 ∈ ω ∧ 𝑦 = (𝐴 ↑o 𝑥)) ↔ 𝑧 ∈ ω)) |
| 70 | 29, 69 | bitrid 283 |
. . . 4
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (𝑧 ∈ ∪ {𝑦
∣ ∃𝑥 ∈
ω 𝑦 = (𝐴 ↑o 𝑥)} ↔ 𝑧 ∈ ω)) |
| 71 | 70 | eqrdv 2735 |
. . 3
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ ∪ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴 ↑o 𝑥)} = ω) |
| 72 | 19, 71 | eqtrid 2789 |
. 2
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ ∪ 𝑥 ∈ ω (𝐴 ↑o 𝑥) = ω) |
| 73 | 17, 72 | eqtrd 2777 |
1
⊢ ((𝐴 ∈ ω ∧
1o ∈ 𝐴)
→ (𝐴
↑o ω) = ω) |