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Mirrors > Home > MPE Home > Th. List > Mathboxes > axnulg | Structured version Visualization version GIF version |
Description: A generalization of ax-nul 5312 in which 𝑥 and 𝑦 need not be distinct. Note that it is possible to use axc7e 2317 to derive elirrv 9634 from this theorem, which justifies the dependency on ax-reg 9630. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axnulg | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnae 2437 | . . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | nfcvf 2930 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
3 | nfcvd 2904 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧) | |
4 | 2, 3 | nfeld 2915 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
5 | 4 | nfnd 1856 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 ¬ 𝑦 ∈ 𝑧) |
6 | 1, 5 | nfald 2327 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦 ¬ 𝑦 ∈ 𝑧) |
7 | nfvd 1913 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
8 | dveeq2 2381 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥)) | |
9 | 8 | naecoms 2432 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥)) |
10 | elequ2 2121 | . . . . . 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 1812 | . . 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 2113 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
16 | 15 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥)) |
17 | 16 | sps 2183 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥)) |
18 | 17 | dral1 2442 | . . 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 5312 | . 2 ⊢ ∃𝑧∀𝑦 ¬ 𝑦 ∈ 𝑧 | |
21 | elirrv 9634 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
22 | 21 | ax-gen 1792 | . . 3 ⊢ ∀𝑥 ¬ 𝑥 ∈ 𝑥 |
23 | 22 | exgen 1972 | . 2 ⊢ ∃𝑥∀𝑥 ¬ 𝑥 ∈ 𝑥 |
24 | 6, 7, 14, 19, 20, 23 | dvelimexcasei 35071 | 1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1535 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: (None) |
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