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Mirrors > Home > ILE Home > Th. List > neneqad | GIF version |
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
neneqad.1 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
neneqad | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neneqad.1 | . . 3 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
2 | 1 | con2i 557 | . 2 ⊢ (𝐴 = 𝐵 → ¬ 𝜑) |
3 | 2 | necon2ai 2259 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 df-ne 2206 |
This theorem is referenced by: ne0i 3230 nsuceq0g 4155 fidifsnen 6331 nqnq0pi 6536 xrlttri3 8718 expival 9257 |
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