Theorem xpnz 5553

Description: The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)

Assertion

Ref	Expression
xpnz	|- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )

Proof of Theorem xpnz

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . . . 5 |- ( A =/= (/) <-> E. x x e. A )
2		n0 3931	. . . . 5 |- ( B =/= (/) <-> E. y y e. B )
3	1, 2	anbi12i 733	. . . 4 |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( E. x x e. A /\ E. y y e. B ) )
4		eeanv 2182	. . . 4 |- ( E. x E. y ( x e. A /\ y e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
5	3, 4	bitr4i 267	. . 3 |- ( ( A =/= (/) /\ B =/= (/) ) <-> E. x E. y ( x e. A /\ y e. B ) )
6		opex 4932	. . . . . 6 |- <. x , y >. e. _V
7		eleq1 2689	. . . . . . 7 |- ( z = <. x , y >. -> ( z e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
8		opelxp 5146	. . . . . . 7 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
9	7, 8	syl6bb 276	. . . . . 6 |- ( z = <. x , y >. -> ( z e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
10	6, 9	spcev 3300	. . . . 5 |- ( ( x e. A /\ y e. B ) -> E. z z e. ( A X. B ) )
11		n0 3931	. . . . 5 |- ( ( A X. B ) =/= (/) <-> E. z z e. ( A X. B ) )
12	10, 11	sylibr 224	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( A X. B ) =/= (/) )
13	12	exlimivv 1860	. . 3 |- ( E. x E. y ( x e. A /\ y e. B ) -> ( A X. B ) =/= (/) )
14	5, 13	sylbi 207	. 2 |- ( ( A =/= (/) /\ B =/= (/) ) -> ( A X. B ) =/= (/) )
15		xpeq1 5128	. . . . 5 |- ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
16		0xp 5199	. . . . 5 |- ( (/) X. B ) = (/)
17	15, 16	syl6eq 2672	. . . 4 |- ( A = (/) -> ( A X. B ) = (/) )
18	17	necon3i 2826	. . 3 |- ( ( A X. B ) =/= (/) -> A =/= (/) )
19		xpeq2 5129	. . . . 5 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
20		xp0 5552	. . . . 5 |- ( A X. (/) ) = (/)
21	19, 20	syl6eq 2672	. . . 4 |- ( B = (/) -> ( A X. B ) = (/) )
22	21	necon3i 2826	. . 3 |- ( ( A X. B ) =/= (/) -> B =/= (/) )
23	18, 22	jca 554	. 2 |- ( ( A X. B ) =/= (/) -> ( A =/= (/) /\ B =/= (/) ) )
24	14, 23	impbii 199	1 |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  xpeq0  5554  ssxpb  5568  xp11  5569  unixpid  5670  xpexr2  7107  frxp  7287  xpfir  8182  axcc2lem  9258  axdc4lem  9277  mamufacex  20195  txindis  21437  bj-xpnzex  32946  bj-1upln0  32997  bj-2upln1upl  33012  dibn0  36442