Theorem opelxp2 4675

Description: The second member of an ordered pair of classes in a cross product belongs to second cross 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

Step	Hyp	Ref	Expression
1		opelxp 4670	. 2 |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
2	1	simprbi 275	1 |- ( <. A , B >. e. ( C X. D ) -> B e. D )

Syntax hints:   -> wi 4   e. wcel 2159   <.cop 3609   X. cxp 4638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-opab 4079  df-xp 4646

This theorem is referenced by:  dff4im  5677  txlm  14162