Theorem xpm 5032

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 5031	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. w w e. ( A X. B ) )
2		eleq1 2233	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
3	2	cbvexv 1911	. . 3 |- ( E. a a e. A <-> E. x x e. A )
4		eleq1 2233	. . . 4 |- ( b = y -> ( b e. B <-> y e. B ) )
5	4	cbvexv 1911	. . 3 |- ( E. b b e. B <-> E. y y e. B )
6	3, 5	anbi12i 457	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
7		eleq1 2233	. . 3 |- ( w = z -> ( w e. ( A X. B ) <-> z e. ( A X. B ) ) )
8	7	cbvexv 1911	. 2 |- ( E. w w e. ( A X. B ) <-> E. z z e. ( A X. B ) )
9	1, 6, 8	3bitr3i 209	1 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1485   e. wcel 2141   X. cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617

This theorem is referenced by:  ssxpbm  5046  xp11m  5049  xpexr2m  5052  unixpm  5146