Theorem elxpi 4675

Description: Membership in a cross product. Uses fewer axioms than elxp 4676. (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 2200	. . . . . 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 1879	. . . 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 2906	. . 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 4665	. . 3 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
7		df-opab 4091	. . 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 2214	. 2 |- ( B X. C ) = { z | E. x E. y ( z = <. x , y >. /\ ( x e. B /\ y e. C ) ) }
9	5, 8	eleq2s 2288	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 1364   E.wex 1503   e. wcel 2164   {cab 2179   <.cop 3621   {copab 4089   X. cxp 4657

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-opab 4091  df-xp 4665

This theorem is referenced by:  xpsspw  4771  dmaddpqlem  7437  nqpi  7438  enq0ref  7493  nqnq0  7501  nq0nn  7502  cnm  7892  axaddcl  7924  axmulcl  7926