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Mirrors > Home > ILE Home > Th. List > neqne | GIF version |
Description: From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
neqne | ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝐴 = 𝐵) | |
2 | 1 | neqned 2341 | 1 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1342 ≠ wne 2334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-ne 2335 |
This theorem is referenced by: xaddf 9772 seq3f1olemstep 10427 dvdsabseq 11771 |
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