Theorem elxpi 4644

Description: Membership in a cross product. Uses fewer axioms than elxp 4645. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
elxpi	|- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem elxpi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2184	. . . . . 6 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
2	1	anbi1d 465	. . . . 5 |- ( z = A -> ( ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
3	2	2exbidv 1868	. . . 4 |- ( z = A -> ( E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
4	3	elabg 2885	. . 3 |- ( A e. { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) } -> ( A e. { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) } <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
5	4	ibi 176	. 2 |- ( A e. { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) } -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
6		df-xp 4634	. . 3 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
7		df-opab 4067	. . 3 |- { <. x , y >. | ( x e. B /\ y e. C ) } = { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) }
8	6, 7	eqtri 2198	. 2 |- ( B X. C ) = { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) }
9	5, 8	eleq2s 2272	1 |- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   <.cop 3597   {copab 4065   X. cxp 4626

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-opab 4067  df-xp 4634

This theorem is referenced by:  xpsspw  4740  dmaddpqlem  7378  nqpi  7379  enq0ref  7434  nqnq0  7442  nq0nn  7443  cnm  7833  axaddcl  7865  axmulcl  7867