Theorem otelxp1 4583

Description: The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)

Assertion

Ref	Expression
otelxp1	|- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> A e. R )

Proof of Theorem otelxp1

Step	Hyp	Ref	Expression
1		opelxp1 4581	. 2 |- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> <. A , B >. e. ( R X. S ) )
2		opelxp1 4581	. 2 |- ( <. A , B >. e. ( R X. S ) -> A e. R )
3	1, 2	syl 14	1 |- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> A e. R )

Syntax hints:   -> wi 4   e. wcel 1481   <.cop 3535   X. cxp 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553

This theorem is referenced by: (None)