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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 5475 | . 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴)) |
4 | eqcom 2739 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | anbi2i 623 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)) |
6 | anidm 565 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵) | |
7 | 3, 5, 6 | 3bitri 296 | 1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 |
This theorem is referenced by: (None) |
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