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Theorem opelxp2 5673
Description: The second member of an ordered pair of classes in a Cartesian product belongs to second Cartesian product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelxp2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)

Proof of Theorem opelxp2
StepHypRef Expression
1 opelxp 5667 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
21simprbi 498 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cop 4591   × cxp 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167  df-xp 5637
This theorem is referenced by:  dff4  7046  eceqoveq  8695  axdc4lem  10325  canthp1lem2  10523  cicrcl  17622  txcmplem1  22920  txlm  22927  brcgr  27654  nvex  29358  fldextfld2  32129  prsrn  32276  pprodss4v  34400  poimirlem27  36036  natglobalincr  44907
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