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Theorem opelxp2 5731
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 5724 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
21simprbi 496 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cop 4636   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5210  df-xp 5694
This theorem is referenced by:  dff4  7120  eceqoveq  8860  axdc4lem  10492  canthp1lem2  10690  cicrcl  17850  txcmplem1  23664  txlm  23671  brcgr  28929  nvex  30639  fldextfld2  33677  prsrn  33875  pprodss4v  35865  poimirlem27  37633  natglobalincr  46830
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