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Mirrors > Home > MPE Home > Th. List > opcom | Structured version Visualization version 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 5370 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴)) |
4 | eqcom 2830 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | anbi2i 624 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)) |
6 | anidm 567 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵) | |
7 | 3, 5, 6 | 3bitri 299 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: (None) |
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