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Mirrors > Home > ILE Home > Th. List > neqned | 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.) Allow a shortening of necon3bi 2255. (Revised by Wolf Lammen, 22-Nov-2019.) |
Ref | Expression |
---|---|
neqned.1 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
neqned | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neqned.1 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
2 | df-ne 2206 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 1, 2 | sylibr 137 | 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 |
This theorem depends on definitions: df-bi 110 df-ne 2206 |
This theorem is referenced by: neqne 2214 flqltnz 9129 |
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