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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 1551 | . 2 ⊢ ¬ ⊥ | |
2 | eqneqall 3027 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ⊥)) | |
3 | 2 | imp 409 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) → ⊥) |
4 | 1, 3 | mto 199 | 1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ⊥wfal 1549 ≠ wne 3016 |
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 209 df-an 399 df-tru 1540 df-fal 1550 df-ne 3017 |
This theorem is referenced by: osumcllem11N 37117 pexmidlem8N 37128 dochexmidlem8 38618 |
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