Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2157 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | notnoti 635 | . 2 ⊢ ¬ ¬ 𝐴 = 𝐴 |
3 | df-ne 2328 | . . 3 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴) | |
4 | 3 | notbii 658 | . 2 ⊢ (¬ 𝐴 ≠ 𝐴 ↔ ¬ ¬ 𝐴 = 𝐴) |
5 | 2, 4 | mpbir 145 | 1 ⊢ ¬ 𝐴 ≠ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1335 ≠ wne 2327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-gen 1429 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-cleq 2150 df-ne 2328 |
This theorem is referenced by: neldifsn 3691 0nnq 7286 1nuz2 9522 |
Copyright terms: Public domain | W3C validator |