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Mirrors > Home > ILE Home > Th. List > nonconne | GIF version |
Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) |
Ref | Expression |
---|---|
nonconne | ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.24 662 | . 2 ⊢ ¬ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵) | |
2 | df-ne 2256 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | anbi2i 445 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵)) |
4 | 1, 3 | mtbir 631 | 1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 = wceq 1289 ≠ wne 2255 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 |
This theorem depends on definitions: df-bi 115 df-ne 2256 |
This theorem is referenced by: (None) |
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