| Mathbox for BTernaryTau |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > axnulg | Structured version Visualization version GIF version | ||
| Description: A generalization of ax-nul 5306 in which 𝑥 and 𝑦 need not be distinct. Note that it is possible to use axc7e 2318 to derive elirrv 9636 from this theorem, which justifies the dependency on ax-reg 9632. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| axnulg | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfnae 2439 | . . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 2 | nfcvf 2932 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
| 3 | nfcvd 2906 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧) | |
| 4 | 2, 3 | nfeld 2917 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
| 5 | 4 | nfnd 1858 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 ¬ 𝑦 ∈ 𝑧) |
| 6 | 1, 5 | nfald 2328 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦 ¬ 𝑦 ∈ 𝑧) |
| 7 | nfvd 1915 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
| 8 | dveeq2 2383 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥)) | |
| 9 | 8 | naecoms 2434 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥)) |
| 10 | elequ2 2123 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥)) | |
| 11 | 10 | notbid 318 | . . . . 5 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑥)) |
| 12 | 11 | biimpd 229 | . . . 4 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑥)) |
| 13 | 12 | al2imi 1815 | . . 3 ⊢ (∀𝑦 𝑧 = 𝑥 → (∀𝑦 ¬ 𝑦 ∈ 𝑧 → ∀𝑦 ¬ 𝑦 ∈ 𝑥)) |
| 14 | 9, 13 | syl6 35 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (∀𝑦 ¬ 𝑦 ∈ 𝑧 → ∀𝑦 ¬ 𝑦 ∈ 𝑥))) |
| 15 | elequ1 2115 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
| 16 | 15 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥)) |
| 17 | 16 | sps 2185 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥)) |
| 18 | 17 | dral1 2444 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)) |
| 19 | 18 | biimpd 229 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ¬ 𝑦 ∈ 𝑥)) |
| 20 | ax-nul 5306 | . 2 ⊢ ∃𝑧∀𝑦 ¬ 𝑦 ∈ 𝑧 | |
| 21 | elirrv 9636 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
| 22 | 21 | ax-gen 1795 | . . 3 ⊢ ∀𝑥 ¬ 𝑥 ∈ 𝑥 |
| 23 | 22 | exgen 1974 | . 2 ⊢ ∃𝑥∀𝑥 ¬ 𝑥 ∈ 𝑥 |
| 24 | 6, 7, 14, 19, 20, 23 | dvelimexcasei 35092 | 1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |