Theorem xpnz 5046

Description: The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by set.mm contributors, 30-Jun-2006.) (Revised by set.mm contributors, 19-Apr-2007.)

Assertion

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

Proof of Theorem xpnz

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

Step	Hyp	Ref	Expression
1		n0 3560	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
2		n0 3560	. . . . 5 |- (B =/= (/) <-> E.y y e. B)
3	1, 2	anbi12i 678	. . . 4 |- ((A =/= (/) /\ B =/= (/)) <-> (E.x x e. A /\ E.y y e. B))
4		eeanv 1913	. . . 4 |- (E.xE.y(x e. A /\ y e. B) <-> (E.x x e. A /\ E.y y e. B))
5	3, 4	bitr4i 243	. . 3 |- ((A =/= (/) /\ B =/= (/)) <-> E.xE.y(x e. A /\ y e. B))
6		opelxp 4812	. . . . 5 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
7		ne0i 3557	. . . . 5 |- (<.x, y>. e. (A X. B) -> (A X. B) =/= (/))
8	6, 7	sylbir 204	. . . 4 |- ((x e. A /\ y e. B) -> (A X. B) =/= (/))
9	8	exlimivv 1635	. . 3 |- (E.xE.y(x e. A /\ y e. B) -> (A X. B) =/= (/))
10	5, 9	sylbi 187	. 2 |- ((A =/= (/) /\ B =/= (/)) -> (A X. B) =/= (/))
11		xpeq1 4799	. . . . 5 |- (A = (/) -> (A X. B) = ((/) X. B))
12		xp0r 4843	. . . . 5 |- ((/) X. B) = (/)
13	11, 12	syl6eq 2401	. . . 4 |- (A = (/) -> (A X. B) = (/))
14	13	necon3i 2556	. . 3 |- ((A X. B) =/= (/) -> A =/= (/))
15		xpeq2 4800	. . . . 5 |- (B = (/) -> (A X. B) = (A X. (/)))
16		xp0 5045	. . . . 5 |- (A X. (/)) = (/)
17	15, 16	syl6eq 2401	. . . 4 |- (B = (/) -> (A X. B) = (/))
18	17	necon3i 2556	. . 3 |- ((A X. B) =/= (/) -> B =/= (/))
19	14, 18	jca 518	. 2 |- ((A X. B) =/= (/) -> (A =/= (/) /\ B =/= (/)))
20	10, 19	impbii 180	1 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  (/)c0 3551  <.cop 4562   X. cxp 4771

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-xp 4785  df-cnv 4786

This theorem is referenced by:  xpeq0  5047  ssxpb  5056  xp11  5057  xpexr2  5111