Theorem elxpi 4627

Description: Membership in a cross product. Uses fewer axioms than elxp 4628. (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 2177	. . . . . 6 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
2	1	anbi1d 462	. . . . 5 |- ( z = A -> ( ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
3	2	2exbidv 1861	. . . 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 2876	. . 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 175	. 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 4617	. . 3 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
7		df-opab 4051	. . 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 2191	. 2 |- ( B X. C ) = { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) }
9	5, 8	eleq2s 2265	1 |- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   <.cop 3586   {copab 4049   X. cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-opab 4051  df-xp 4617

This theorem is referenced by:  xpsspw  4723  dmaddpqlem  7339  nqpi  7340  enq0ref  7395  nqnq0  7403  nq0nn  7404  cnm  7794  axaddcl  7826  axmulcl  7828