Step | Hyp | Ref
| Expression |
1 | | unfilem1.2 |
. . . . . . . . . 10
⊢ 𝐵 ∈ ω |
2 | | elnn 7715 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑥 ∈ ω) |
3 | 1, 2 | mpan2 688 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ω) |
4 | | unfilem1.1 |
. . . . . . . . . 10
⊢ 𝐴 ∈ ω |
5 | | nnaord 8433 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))) |
6 | 1, 4, 5 | mp3an23 1452 |
. . . . . . . . 9
⊢ (𝑥 ∈ ω → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))) |
7 | 3, 6 | syl 17 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))) |
8 | 7 | ibi 266 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐵 → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)) |
9 | | nnaword1 8443 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥)) |
10 | | nnord 7712 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ω → Ord 𝐴) |
11 | | nnacl 8425 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈
ω) |
12 | | nnord 7712 |
. . . . . . . . . . 11
⊢ ((𝐴 +o 𝑥) ∈ ω → Ord
(𝐴 +o 𝑥)) |
13 | 11, 12 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord
(𝐴 +o 𝑥)) |
14 | | ordtri1 6297 |
. . . . . . . . . 10
⊢ ((Ord
𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
15 | 10, 13, 14 | syl2an2r 682 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
16 | 9, 15 | mpbid 231 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬
(𝐴 +o 𝑥) ∈ 𝐴) |
17 | 4, 3, 16 | sylancr 587 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴) |
18 | 8, 17 | jca 512 |
. . . . . 6
⊢ (𝑥 ∈ 𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
19 | | eleq1 2828 |
. . . . . . . 8
⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))) |
20 | | eleq1 2828 |
. . . . . . . . 9
⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ 𝐴 ↔ (𝐴 +o 𝑥) ∈ 𝐴)) |
21 | 20 | notbid 318 |
. . . . . . . 8
⊢ (𝑦 = (𝐴 +o 𝑥) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
22 | 19, 21 | anbi12d 631 |
. . . . . . 7
⊢ (𝑦 = (𝐴 +o 𝑥) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))) |
23 | 22 | biimparc 480 |
. . . . . 6
⊢ ((((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) |
24 | 18, 23 | sylan 580 |
. . . . 5
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) |
25 | 24 | rexlimiva 3212 |
. . . 4
⊢
(∃𝑥 ∈
𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) |
26 | 4, 1 | nnacli 8428 |
. . . . . . . 8
⊢ (𝐴 +o 𝐵) ∈ ω |
27 | | elnn 7715 |
. . . . . . . 8
⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω) |
28 | 26, 27 | mpan2 688 |
. . . . . . 7
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω) |
29 | | nnord 7712 |
. . . . . . . . 9
⊢ (𝑦 ∈ ω → Ord 𝑦) |
30 | | ordtri1 6297 |
. . . . . . . . 9
⊢ ((Ord
𝐴 ∧ Ord 𝑦) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴)) |
31 | 10, 29, 30 | syl2an 596 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴)) |
32 | | nnawordex 8451 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦)) |
33 | 31, 32 | bitr3d 280 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬
𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦)) |
34 | 4, 28, 33 | sylancr 587 |
. . . . . 6
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦)) |
35 | | eleq1 2828 |
. . . . . . . . . 10
⊢ ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵))) |
36 | 6, 35 | sylan9bb 510 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ (𝐴 +o 𝐵))) |
37 | 36 | biimprcd 249 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑥 ∈ 𝐵)) |
38 | | eqcom 2747 |
. . . . . . . . . 10
⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥)) |
39 | 38 | biimpi 215 |
. . . . . . . . 9
⊢ ((𝐴 +o 𝑥) = 𝑦 → 𝑦 = (𝐴 +o 𝑥)) |
40 | 39 | adantl 482 |
. . . . . . . 8
⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥)) |
41 | 37, 40 | jca2 514 |
. . . . . . 7
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥)))) |
42 | 41 | reximdv2 3201 |
. . . . . 6
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))) |
43 | 34, 42 | sylbid 239 |
. . . . 5
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))) |
44 | 43 | imp 407 |
. . . 4
⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)) |
45 | 25, 44 | impbii 208 |
. . 3
⊢
(∃𝑥 ∈
𝐵 𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) |
46 | | unfilem1.3 |
. . . 4
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) |
47 | | ovex 7302 |
. . . 4
⊢ (𝐴 +o 𝑥) ∈ V |
48 | 46, 47 | elrnmpti 5867 |
. . 3
⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)) |
49 | | eldif 3902 |
. . 3
⊢ (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) |
50 | 45, 48, 49 | 3bitr4i 303 |
. 2
⊢ (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) |
51 | 50 | eqriv 2737 |
1
⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴) |