Theorem elxpi 4379

Description: Membership in a cross product. Uses fewer axioms than elxp 4380. (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 2087	. . . . . 6 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
2	1	anbi1d 452	. . . . 5 |- ( z = A -> ( ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
3	2	2exbidv 1789	. . . 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 2739	. . 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 174	. 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 4369	. . 3 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
7		df-opab 3840	. . 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 2101	. 2 |- ( B X. C ) = { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) }
9	5, 8	eleq2s 2173	1 |- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   <.cop 3401   {copab 3838   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-opab 3840  df-xp 4369

This theorem is referenced by:  xpsspw  4468  dmaddpqlem  6567  nqpi  6568  enq0ref  6623  nqnq0  6631  nq0nn  6632  axaddcl  7032  axmulcl  7034