Theorem xpnz 3466

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 2288	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
2		ne0 2288	. . . . 5 |- (B =/= (/) <-> E.y y e. B)
3	1, 2	anbi12i 482	. . . 4 |- ((A =/= (/) /\ B =/= (/)) <-> (E.x x e. A /\ E.y y e. B))
4		eeanv 1323	. . . 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 2782	. . . . . 6 |- <.x, y>. e. V
7		eleq1 1534	. . . . . . 7 |- (z = <.x, y>. -> (z e. (A X. B) <-> <.x, y>. e. (A X. B)))
8		visset 1813	. . . . . . . 8 |- y e. V
9	8	opelxp 3214	. . . . . . 7 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
10	7, 9	syl6bb 536	. . . . . 6 |- (z = <.x, y>. -> (z e. (A X. B) <-> (x e. A /\ y e. B)))
11	6, 10	cla4ev 1869	. . . . 5 |- ((x e. A /\ y e. B) -> E.z z e. (A X. B))
12		ne0 2288	. . . . 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 1296	. . 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 3200	. . . . 5 |- (A = (/) -> (A X. B) = ((/) X. B))
17		xp0r 3239	. . . . 5 |- ((/) X. B) = (/)
18	16, 17	syl6eq 1523	. . . 4 |- (A = (/) -> (A X. B) = (/))
19	18	necon3i 1605	. . 3 |- ((A X. B) =/= (/) -> A =/= (/))
20		xpeq2 3201	. . . . 5 |- (B = (/) -> (A X. B) = (A X. (/)))
21		xp0 3465	. . . . 5 |- (A X. (/)) = (/)
22	20, 21	syl6eq 1523	. . . 4 |- (B = (/) -> (A X. B) = (/))
23	22	necon3i 1605	. . 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 956   e. wcel 958  E.wex 980   =/= wne 1585  (/)c0 2280  <.cop 2411   X. cxp 3168

This theorem is referenced by:  xpeq0 3467  ssxpr 3475  xp11 3476  xpexr2 3480  relrded 10675  relrcat 10696

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186

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