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Theorem opelxp2 4873
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 >.  e.  ( C  X.  D
)  ->  B  e.  D )

Proof of Theorem opelxp2
StepHypRef Expression
1 opelxp 4869 . 2  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
21simprbi 471 1  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  B  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1904   <.cop 3965    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845
This theorem is referenced by:  dff4  6051  eceqoveq  7486  isfin4-3  8763  axdc4lem  8903  canthp1lem2  9096  cicrcl  15786  txcmplem1  20733  txlm  20740  brcgr  25009  nvex  26311  prsrn  28795  pprodss4v  30722  poimirlem27  32031
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