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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 5589 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ (V × V) |
4 | eleq1 2900 | . . 3 ⊢ (𝐴 = 〈𝐵, 𝐶〉 → (𝐴 ∈ (V × V) ↔ 〈𝐵, 𝐶〉 ∈ (V × V))) | |
5 | 3, 4 | mpbiri 260 | . 2 ⊢ (𝐴 = 〈𝐵, 𝐶〉 → 𝐴 ∈ (V × V)) |
6 | eqop 7725 | . 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | |
7 | 5, 6 | biadanii 820 | 1 ⊢ (𝐴 = 〈𝐵, 𝐶〉 ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 〈cop 4567 × cxp 5548 ‘cfv 6350 1st c1st 7681 2nd c2nd 7682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fv 6358 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: evlslem4 20282 |
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