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Theorem xpopth 5725
Description: An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
Assertion
Ref Expression
xpopth ((A (𝐶 × 𝐷) B (𝑅 × 𝑆)) → (((1stA) = (1stB) (2ndA) = (2ndB)) ↔ A = B))

Proof of Theorem xpopth
StepHypRef Expression
1 1st2nd2 5724 . . 3 (A (𝐶 × 𝐷) → A = ⟨(1stA), (2ndA)⟩)
2 1st2nd2 5724 . . 3 (B (𝑅 × 𝑆) → B = ⟨(1stB), (2ndB)⟩)
31, 2eqeqan12d 2037 . 2 ((A (𝐶 × 𝐷) B (𝑅 × 𝑆)) → (A = B ↔ ⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩))
4 1stexg 5717 . . . 4 (A (𝐶 × 𝐷) → (1stA) V)
5 2ndexg 5718 . . . 4 (A (𝐶 × 𝐷) → (2ndA) V)
6 opthg 3949 . . . 4 (((1stA) V (2ndA) V) → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
74, 5, 6syl2anc 393 . . 3 (A (𝐶 × 𝐷) → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
87adantr 261 . 2 ((A (𝐶 × 𝐷) B (𝑅 × 𝑆)) → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
93, 8bitr2d 178 1 ((A (𝐶 × 𝐷) B (𝑅 × 𝑆)) → (((1stA) = (1stB) (2ndA) = (2ndB)) ↔ A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  cop 3353   × cxp 4270  cfv 4829  1st c1st 5688  2nd c2nd 5689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by: (None)
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