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Theorem opelxp2 5565
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 5559 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
21simprbi 500 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cop 4534   × cxp 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529
This theorem is referenced by:  dff4  6848  eceqoveq  8389  axdc4lem  9870  canthp1lem2  10068  cicrcl  17069  txcmplem1  22250  txlm  22257  brcgr  26698  nvex  28398  fldextfld2  31132  prsrn  31272  pprodss4v  33459  poimirlem27  35083
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