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Mirrors > Home > ILE Home > Th. List > nner | GIF version |
Description: Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.) |
Ref | Expression |
---|---|
nner | ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2346 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | 1 | biimpi 120 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) |
3 | 2 | con2i 627 | 1 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1353 ≠ wne 2345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-in1 614 ax-in2 615 |
This theorem depends on definitions: df-bi 117 df-ne 2346 |
This theorem is referenced by: nn0eln0 4613 funtpg 5259 fin0 6875 hashnncl 10742 |
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