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Mirrors > Home > ILE Home > Th. List > eqop2 | 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 4408 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ (V × V) |
4 | eleq1 2141 | . . 3 ⊢ (𝐴 = 〈𝐵, 𝐶〉 → (𝐴 ∈ (V × V) ↔ 〈𝐵, 𝐶〉 ∈ (V × V))) | |
5 | 3, 4 | mpbiri 166 | . 2 ⊢ (𝐴 = 〈𝐵, 𝐶〉 → 𝐴 ∈ (V × V)) |
6 | eqop 5823 | . 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | |
7 | 5, 6 | biadan2 443 | 1 ⊢ (𝐴 = 〈𝐵, 𝐶〉 ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 Vcvv 2601 〈cop 3401 × cxp 4361 ‘cfv 4922 1st c1st 5785 2nd c2nd 5786 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-1st 5787 df-2nd 5788 |
This theorem is referenced by: (None) |
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