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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 2948 | . 2 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
| 2 | 1 | orri 863 | 1 ⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ∨ wo 848 = wceq 1540 ≠ wne 2940 |
| 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 207 df-or 849 df-ne 2941 |
| This theorem is referenced by: elnn1uz2 12967 hashv01gt1 14384 numclwwlk3lem2lem 30402 hashxpe 32811 drngmxidlr 33506 constrfin 33787 constrelextdg2 33788 subfacp1lem6 35190 tendoeq2 40776 ax6e2ndeqVD 44929 ax6e2ndeqALT 44951 |
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