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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 4064 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴)) |
4 | eqcom 2090 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | anbi2i 445 | . 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 1289 ∈ wcel 1438 Vcvv 2619 〈cop 3449 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 |
This theorem is referenced by: (None) |
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