Theorem elxpi 4550

Description: Membership in a cross product. Uses fewer axioms than elxp 4551. (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 2144	. . . . . 6 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
2	1	anbi1d 460	. . . . 5 |- ( z = A -> ( ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
3	2	2exbidv 1840	. . . 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 2825	. . 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 4540	. . 3 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
7		df-opab 3985	. . 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 2158	. 2 |- ( B X. C ) = { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) }
9	5, 8	eleq2s 2232	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 1331   E.wex 1468   e. wcel 1480   {cab 2123   <.cop 3525   {copab 3983   X. cxp 4532

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-opab 3985  df-xp 4540

This theorem is referenced by:  xpsspw  4646  dmaddpqlem  7178  nqpi  7179  enq0ref  7234  nqnq0  7242  nq0nn  7243  cnm  7633  axaddcl  7665  axmulcl  7667