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Mirrors > Home > MPE Home > Th. List > eqop2 | Structured version Visualization version GIF version |
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
Ref | Expression |
---|---|
eqop2.1 | ⊢ 𝐵 ∈ V |
eqop2.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
eqop2 | ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqop2.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | eqop2.2 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | 1, 2 | opelvv 5712 | . . 3 ⊢ ⟨𝐵, 𝐶⟩ ∈ (V × V) |
4 | eleq1 2817 | . . 3 ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ → (𝐴 ∈ (V × V) ↔ ⟨𝐵, 𝐶⟩ ∈ (V × V))) | |
5 | 3, 4 | mpbiri 258 | . 2 ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ → 𝐴 ∈ (V × V)) |
6 | eqop 8029 | . 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | |
7 | 5, 6 | biadanii 821 | 1 ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⟨cop 4630 × cxp 5670 ‘cfv 6542 1st c1st 7985 2nd c2nd 7986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fv 6550 df-1st 7987 df-2nd 7988 |
This theorem is referenced by: evlslem4 22013 |
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