| Step | Hyp | Ref
| Expression |
| 1 | | unfilem1.2 |
. . . . . . . . . 10
⊢ 𝐵 ∈ ω |
| 2 | | elnn 7898 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑥 ∈ ω) |
| 3 | 1, 2 | mpan2 691 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ω) |
| 4 | | unfilem1.1 |
. . . . . . . . . 10
⊢ 𝐴 ∈ ω |
| 5 | | nnaord 8657 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))) |
| 6 | 1, 4, 5 | mp3an23 1455 |
. . . . . . . . 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 267 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐵 → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)) |
| 9 | | nnaword1 8667 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥)) |
| 10 | | nnord 7895 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ω → Ord 𝐴) |
| 11 | | nnacl 8649 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈
ω) |
| 12 | | nnord 7895 |
. . . . . . . . . . 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 6417 |
. . . . . . . . . 10
⊢ ((Ord
𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
| 15 | 10, 13, 14 | syl2an2r 685 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
| 16 | 9, 15 | mpbid 232 |
. . . . . . . 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 511 |
. . . . . 6
⊢ (𝑥 ∈ 𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴)) |
| 19 | | eleq1 2829 |
. . . . . . . 8
⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))) |
| 20 | | eleq1 2829 |
. . . . . . . . 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 632 |
. . . . . . 7
⊢ (𝑦 = (𝐴 +o 𝑥) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))) |
| 23 | 22 | biimparc 479 |
. . . . . 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 3147 |
. . . 4
⊢
(∃𝑥 ∈
𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) |
| 26 | 4, 1 | nnacli 8652 |
. . . . . . . 8
⊢ (𝐴 +o 𝐵) ∈ ω |
| 27 | | elnn 7898 |
. . . . . . . 8
⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω) |
| 28 | 26, 27 | mpan2 691 |
. . . . . . 7
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω) |
| 29 | | nnord 7895 |
. . . . . . . . 9
⊢ (𝑦 ∈ ω → Ord 𝑦) |
| 30 | | ordtri1 6417 |
. . . . . . . . 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 8675 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦)) |
| 33 | 31, 32 | bitr3d 281 |
. . . . . . 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 2829 |
. . . . . . . . . 10
⊢ ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵))) |
| 36 | 6, 35 | sylan9bb 509 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ (𝐴 +o 𝐵))) |
| 37 | 36 | biimprcd 250 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑥 ∈ 𝐵)) |
| 38 | | eqcom 2744 |
. . . . . . . . . 10
⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥)) |
| 39 | 38 | biimpi 216 |
. . . . . . . . 9
⊢ ((𝐴 +o 𝑥) = 𝑦 → 𝑦 = (𝐴 +o 𝑥)) |
| 40 | 39 | adantl 481 |
. . . . . . . 8
⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥)) |
| 41 | 37, 40 | jca2 513 |
. . . . . . 7
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥)))) |
| 42 | 41 | reximdv2 3164 |
. . . . . 6
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))) |
| 43 | 34, 42 | sylbid 240 |
. . . . 5
⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))) |
| 44 | 43 | imp 406 |
. . . 4
⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)) |
| 45 | 25, 44 | impbii 209 |
. . 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 7464 |
. . . 4
⊢ (𝐴 +o 𝑥) ∈ V |
| 48 | 46, 47 | elrnmpti 5973 |
. . 3
⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)) |
| 49 | | eldif 3961 |
. . 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 2734 |
1
⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴) |