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Mirrors > Home > ILE Home > Th. List > opcom | GIF version |
Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) |
Ref | Expression |
---|---|
opcom.1 | ⊢ 𝐴 ∈ V |
opcom.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opcom | ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opcom.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | opcom.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opth 3992 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴)) |
4 | eqcom 2083 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | anbi2i 444 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)) |
6 | anidm 388 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵) | |
7 | 3, 5, 6 | 3bitri 204 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 Vcvv 2601 〈cop 3401 |
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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 |
This theorem is referenced by: (None) |
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