Theorem xpm 4765

Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Assertion

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

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

Allowed substitution hints:   A( y)   B( x)

Proof of Theorem xpm

Dummy variables a b w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpmlem 4764	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. w w e. ( A X. B ) )
2		eleq1 2141	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
3	2	cbvexv 1836	. . 3 |- ( E. a a e. A <-> E. x x e. A )
4		eleq1 2141	. . . 4 |- ( b = y -> ( b e. B <-> y e. B ) )
5	4	cbvexv 1836	. . 3 |- ( E. b b e. B <-> E. y y e. B )
6	3, 5	anbi12i 447	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
7		eleq1 2141	. . 3 |- ( w = z -> ( w e. ( A X. B ) <-> z e. ( A X. B ) ) )
8	7	cbvexv 1836	. 2 |- ( E. w w e. ( A X. B ) <-> E. z z e. ( A X. B ) )
9	1, 6, 8	3bitr3i 208	1 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369

This theorem is referenced by:  ssxpbm  4776  xp11m  4779  xpexr2m  4782  unixpm  4873