Proof of Theorem axprlem5
Step | Hyp | Ref
| Expression |
1 | | ax-nul 5210 |
. 2
⊢
∃𝑠∀𝑛 ¬ 𝑛 ∈ 𝑠 |
2 | | nfa1 2155 |
. . . 4
⊢
Ⅎ𝑠∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) |
3 | | nfv 1915 |
. . . 4
⊢
Ⅎ𝑠 𝑤 = 𝑦 |
4 | 2, 3 | nfan 1900 |
. . 3
⊢
Ⅎ𝑠(∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) |
5 | | pm2.21 123 |
. . . . . . . . 9
⊢ (¬
𝑛 ∈ 𝑠 → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛)) |
6 | 5 | alimi 1812 |
. . . . . . . 8
⊢
(∀𝑛 ¬
𝑛 ∈ 𝑠 → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛)) |
7 | 6 | adantr 483 |
. . . . . . 7
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛)) |
8 | | df-ral 3143 |
. . . . . . 7
⊢
(∀𝑛 ∈
𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 ↔ ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛)) |
9 | 7, 8 | sylibr 236 |
. . . . . 6
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛) |
10 | | sp 2182 |
. . . . . . 7
⊢
(∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)) |
11 | 10 | ad2antrl 726 |
. . . . . 6
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)) |
12 | 9, 11 | mpd 15 |
. . . . 5
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → 𝑠 ∈ 𝑝) |
13 | | simpl 485 |
. . . . . . 7
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ¬ 𝑛 ∈ 𝑠) |
14 | | alnex 1782 |
. . . . . . 7
⊢
(∀𝑛 ¬
𝑛 ∈ 𝑠 ↔ ¬ ∃𝑛 𝑛 ∈ 𝑠) |
15 | 13, 14 | sylib 220 |
. . . . . 6
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ¬ ∃𝑛 𝑛 ∈ 𝑠) |
16 | | simprr 771 |
. . . . . 6
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑦) |
17 | | ifpfal 1069 |
. . . . . . 7
⊢ (¬
∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)) |
18 | 17 | biimpar 480 |
. . . . . 6
⊢ ((¬
∃𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑦) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) |
19 | 15, 16, 18 | syl2anc 586 |
. . . . 5
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) |
20 | 12, 19 | jca 514 |
. . . 4
⊢
((∀𝑛 ¬
𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) |
21 | 20 | expcom 416 |
. . 3
⊢
((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → (∀𝑛 ¬ 𝑛 ∈ 𝑠 → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))) |
22 | 4, 21 | eximd 2216 |
. 2
⊢
((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → (∃𝑠∀𝑛 ¬ 𝑛 ∈ 𝑠 → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))) |
23 | 1, 22 | mpi 20 |
1
⊢
((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) |