Theorem xpmlem 5164

Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)

Assertion

Ref	Expression
xpmlem	|- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Distinct variable groups:   x, y, z, A   x, B, y, z

Proof of Theorem xpmlem

Step	Hyp	Ref	Expression
1		eeanv 1985	. . 3 |- ( E. x E. y ( x e. A /\ y e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
2		vex 2806	. . . . . 6 |- x e. _V
3		vex 2806	. . . . . 6 |- y e. _V
4	2, 3	opex 4327	. . . . 5 |- <. x , y >. e. _V
5		eleq1 2294	. . . . . 6 |- ( z = <. x , y >. -> ( z e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
6		opelxp 4761	. . . . . 6 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
7	5, 6	bitrdi 196	. . . . 5 |- ( z = <. x , y >. -> ( z e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
8	4, 7	spcev 2902	. . . 4 |- ( ( x e. A /\ y e. B ) -> E. z z e. ( A X. B ) )
9	8	exlimivv 1945	. . 3 |- ( E. x E. y ( x e. A /\ y e. B ) -> E. z z e. ( A X. B ) )
10	1, 9	sylbir 135	. 2 |- ( ( E. x x e. A /\ E. y y e. B ) -> E. z z e. ( A X. B ) )
11		elxp 4748	. . . . 5 |- ( z e. ( A X. B ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
12		simpr 110	. . . . . 6 |- ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> ( x e. A /\ y e. B ) )
13	12	2eximi 1650	. . . . 5 |- ( E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> E. x E. y ( x e. A /\ y e. B ) )
14	11, 13	sylbi 121	. . . 4 |- ( z e. ( A X. B ) -> E. x E. y ( x e. A /\ y e. B ) )
15	14	exlimiv 1647	. . 3 |- ( E. z z e. ( A X. B ) -> E. x E. y ( x e. A /\ y e. B ) )
16	15, 1	sylib 122	. 2 |- ( E. z z e. ( A X. B ) -> ( E. x x e. A /\ E. y y e. B ) )
17	10, 16	impbii 126	1 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   <.cop 3676   X. cxp 4729

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737

This theorem is referenced by:  xpm  5165