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Mirrors > Home > ILE Home > Th. List > neirr | GIF version |
Description: No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
neirr | ⊢ ¬ 𝐴 ≠ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2083 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | notnoti 607 | . 2 ⊢ ¬ ¬ 𝐴 = 𝐴 |
3 | df-ne 2250 | . . 3 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴) | |
4 | 3 | notbii 627 | . 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ ¬ ¬ 𝐴 = 𝐴) |
5 | 2, 4 | mpbir 144 | 1 ⊢ ¬ 𝐴 ≠ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1285 ≠ wne 2249 |
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 577 ax-in2 578 ax-gen 1379 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-cleq 2076 df-ne 2250 |
This theorem is referenced by: neldifsn 3539 0nnq 6693 1nuz2 8851 |
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