Theorem elxpi 4800

Description: Membership in a cross product. Uses fewer axioms than elxp 4801. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
elxpi	|- (A e. (B X. C) -> E.xE.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 2359	. . . . . 6 |- (z = A -> (z = <.x, y>. <-> A = <.x, y>.))
2	1	anbi1d 685	. . . . 5 |- (z = A -> ((z = <.x, y>. /\ (x e. B /\ y e. C)) <-> (A = <.x, y>. /\ (x e. B /\ y e. C))))
3	2	2exbidv 1628	. . . 4 |- (z = A -> (E.xE.y(z = <.x, y>. /\ (x e. B /\ y e. C)) <-> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C))))
4	3	elabg 2986	. . 3 |- (A e. {z | E.xE.y(z = <.x, y>. /\ (x e. B /\ y e. C))} -> (A e. {z | E.xE.y(z = <.x, y>. /\ (x e. B /\ y e. C))} <-> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C))))
5	4	ibi 232	. 2 |- (A e. {z | E.xE.y(z = <.x, y>. /\ (x e. B /\ y e. C))} -> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))
6		df-xp 4784	. . 3 |- (B X. C) = {<.x, y>. | (x e. B /\ y e. C)}
7		df-opab 4623	. . 3 |- {<.x, y>. | (x e. B /\ y e. C)} = {z | E.xE.y(z = <.x, y>. /\ (x e. B /\ y e. C))}
8	6, 7	eqtri 2373	. 2 |- (B X. C) = {z | E.xE.y(z = <.x, y>. /\ (x e. B /\ y e. C))}
9	5, 8	eleq2s 2445	1 |- (A e. (B X. C) -> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  <.cop 4561  {copab 4622   X. cxp 4770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-opab 4623  df-xp 4784

This theorem is referenced by: (None)