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Theorem xpopth 5744
 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 5743 . . 3 (A (𝐶 × 𝐷) → A = ⟨(1stA), (2ndA)⟩)
2 1st2nd2 5743 . . 3 (B (𝑅 × 𝑆) → B = ⟨(1stB), (2ndB)⟩)
31, 2eqeqan12d 2052 . 2 ((A (𝐶 × 𝐷) B (𝑅 × 𝑆)) → (A = B ↔ ⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩))
4 1stexg 5736 . . . 4 (A (𝐶 × 𝐷) → (1stA) V)
5 2ndexg 5737 . . . 4 (A (𝐶 × 𝐷) → (2ndA) V)
6 opthg 3966 . . . 4 (((1stA) V (2ndA) V) → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
74, 5, 6syl2anc 391 . . 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 1242   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   × cxp 4286  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710 This theorem is referenced by: (None)
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