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Theorem xpopth 8012
Description: An ordered pair theorem for members of Cartesian 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 8010 . . 3 (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 8010 . . 3 (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2eqeqan12d 2740 . 2 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
4 fvex 6897 . . 3 (1st𝐴) ∈ V
5 fvex 6897 . . 3 (2nd𝐴) ∈ V
64, 5opth 5469 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)))
73, 6bitr2di 288 1 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cop 4629   × cxp 5667  cfv 6536  1st c1st 7969  2nd c2nd 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971  df-2nd 7972
This theorem is referenced by:  fseqdom  10020  iundom2g  10534  mdetunilem9  22473  txhaus  23502  fsumvma  27097  wlkeq  29396  disjxpin  32324  poimirlem4  37003  poimirlem13  37012  poimirlem14  37013  poimirlem22  37021  poimirlem26  37025  poimirlem27  37026  rmxypairf1o  42209
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