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Mirrors > Home > MPE Home > Th. List > nonconne | Structured version Visualization version GIF version |
Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.) |
Ref | Expression |
---|---|
nonconne | ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fal 1556 | . 2 ⊢ ¬ ⊥ | |
2 | eqneqall 2952 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ⊥)) | |
3 | 2 | imp 408 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) → ⊥) |
4 | 1, 3 | mto 196 | 1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ⊥wfal 1554 ≠ wne 2941 |
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-an 398 df-tru 1545 df-fal 1555 df-ne 2942 |
This theorem is referenced by: frxp2 8125 osumcllem11N 38775 pexmidlem8N 38786 dochexmidlem8 40276 |
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