Theorem otelxp1 4874

Description: The first member of an ordered triple of classes in a Cartesian product belongs to first Cartesian 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 4872	. 2 |- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> <. A , B >. e. ( R X. S ) )
2		opelxp1 4872	. 2 |- ( <. A , B >. e. ( R X. S ) -> A e. R )
3	1, 2	syl 17	1 |- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> A e. R )

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: (None)