Proof of Theorem fisupg
Step | Hyp | Ref
| Expression |
1 | | fimaxg 8839 |
. 2
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥)) |
2 | | sotrieq2 5472 |
. . . . . . . . . . 11
⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 = 𝑦 ↔ (¬ 𝑥𝑅𝑦 ∧ ¬ 𝑦𝑅𝑥))) |
3 | 2 | simprbda 502 |
. . . . . . . . . 10
⊢ (((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 = 𝑦) → ¬ 𝑥𝑅𝑦) |
4 | 3 | ex 416 |
. . . . . . . . 9
⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 = 𝑦 → ¬ 𝑥𝑅𝑦)) |
5 | 4 | anassrs 471 |
. . . . . . . 8
⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 → ¬ 𝑥𝑅𝑦)) |
6 | 5 | a1dd 50 |
. . . . . . 7
⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 → ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → ¬ 𝑥𝑅𝑦))) |
7 | | pm2.27 42 |
. . . . . . . 8
⊢ (𝑥 ≠ 𝑦 → ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → 𝑦𝑅𝑥)) |
8 | | so2nr 5468 |
. . . . . . . . . 10
⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
9 | | pm3.21 475 |
. . . . . . . . . . 11
⊢ (𝑦𝑅𝑥 → (𝑥𝑅𝑦 → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) |
10 | 9 | con3d 155 |
. . . . . . . . . 10
⊢ (𝑦𝑅𝑥 → (¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → ¬ 𝑥𝑅𝑦)) |
11 | 8, 10 | syl5com 31 |
. . . . . . . . 9
⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑦𝑅𝑥 → ¬ 𝑥𝑅𝑦)) |
12 | 11 | anassrs 471 |
. . . . . . . 8
⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 → ¬ 𝑥𝑅𝑦)) |
13 | 7, 12 | syl9r 78 |
. . . . . . 7
⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → ¬ 𝑥𝑅𝑦))) |
14 | 6, 13 | pm2.61dne 3020 |
. . . . . 6
⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → ¬ 𝑥𝑅𝑦)) |
15 | 14 | ralimdva 3091 |
. . . . 5
⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)) |
16 | | breq2 5034 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥)) |
17 | 16 | rspcev 3526 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) |
18 | 17 | ex 416 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) |
19 | 18 | ralrimivw 3097 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) |
20 | 19 | adantl 485 |
. . . . 5
⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) |
21 | 15, 20 | jctird 530 |
. . . 4
⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)))) |
22 | 21 | reximdva 3184 |
. . 3
⊢ (𝑅 Or 𝐴 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)))) |
23 | 22 | 3ad2ant1 1134 |
. 2
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)))) |
24 | 1, 23 | mpd 15 |
1
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))) |