Theorem xpnz 3472

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

Assertion

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

Proof of Theorem xpnz

Step	Hyp	Ref	Expression
1		ne0 2292	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
2		ne0 2292	. . . . 5 |- (B =/= (/) <-> E.y y e. B)
3	1, 2	anbi12i 484	. . . 4 |- ((A =/= (/) /\ B =/= (/)) <-> (E.x x e. A /\ E.y y e. B))
4		eeanv 1325	. . . 4 |- (E.xE.y(x e. A /\ y e. B) <-> (E.x x e. A /\ E.y y e. B))
5	3, 4	bitr4 176	. . 3 |- ((A =/= (/) /\ B =/= (/)) <-> E.xE.y(x e. A /\ y e. B))
6		opex 2788	. . . . . 6 |- <.x, y>. e. V
7		eleq1 1537	. . . . . . 7 |- (z = <.x, y>. -> (z e. (A X. B) <-> <.x, y>. e. (A X. B)))
8		visset 1816	. . . . . . . 8 |- y e. V
9	8	opelxp 3220	. . . . . . 7 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
10	7, 9	syl6bb 538	. . . . . 6 |- (z = <.x, y>. -> (z e. (A X. B) <-> (x e. A /\ y e. B)))
11	6, 10	cla4ev 1872	. . . . 5 |- ((x e. A /\ y e. B) -> E.z z e. (A X. B))
12		ne0 2292	. . . . 5 |- ((A X. B) =/= (/) <-> E.z z e. (A X. B))
13	11, 12	sylibr 200	. . . 4 |- ((x e. A /\ y e. B) -> (A X. B) =/= (/))
14	13	19.23aivv 1298	. . 3 |- (E.xE.y(x e. A /\ y e. B) -> (A X. B) =/= (/))
15	5, 14	sylbi 199	. 2 |- ((A =/= (/) /\ B =/= (/)) -> (A X. B) =/= (/))
16		xpeq1 3206	. . . . 5 |- (A = (/) -> (A X. B) = ((/) X. B))
17		xp0r 3245	. . . . 5 |- ((/) X. B) = (/)
18	16, 17	syl6eq 1526	. . . 4 |- (A = (/) -> (A X. B) = (/))
19	18	necon3i 1608	. . 3 |- ((A X. B) =/= (/) -> A =/= (/))
20		xpeq2 3207	. . . . 5 |- (B = (/) -> (A X. B) = (A X. (/)))
21		xp0 3471	. . . . 5 |- (A X. (/)) = (/)
22	20, 21	syl6eq 1526	. . . 4 |- (B = (/) -> (A X. B) = (/))
23	22	necon3i 1608	. . 3 |- ((A X. B) =/= (/) -> B =/= (/))
24	19, 23	jca 288	. 2 |- ((A X. B) =/= (/) -> (A =/= (/) /\ B =/= (/)))
25	15, 24	impbi 157	1 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  (/)c0 2283  <.cop 2415   X. cxp 3174

This theorem is referenced by:  xpeq0 3473  ssxpr 3481  xp11 3482  xpexr2 3486  relrded 10646  relrcat 10667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192

Copyright terms: Public domain