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Mirrors > Home > MPE Home > Th. List > exmidne | Structured version Visualization version GIF version |
Description: Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
Ref | Expression |
---|---|
exmidne | ⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neqne 2952 | . 2 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
2 | 1 | orri 861 | 1 ⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 = wceq 1542 ≠ wne 2944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ne 2945 |
This theorem is referenced by: elnn1uz2 12857 hashv01gt1 14252 numclwwlk3lem2lem 29369 hashxpe 31751 subfacp1lem6 33819 tendoeq2 39266 ax6e2ndeqVD 43265 ax6e2ndeqALT 43287 |
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