Theorem unixp0 3504

Description: A cross product is empty iff its union is empty.

Assertion

Ref	Expression
unixp0	|- ((A X. B) = (/) <-> U.(A X. B) = (/))

Proof of Theorem unixp0

Step	Hyp	Ref	Expression
1		unieq 2500	. . 3 |- ((A X. B) = (/) -> U.(A X. B) = U.(/))
2		uni0 2515	. . 3 |- U.(/) = (/)
3	1, 2	syl6eq 1515	. 2 |- ((A X. B) = (/) -> U.(A X. B) = (/))
4		n0 2279	. . . 4 |- (-. (A X. B) = (/) <-> E.z z e. (A X. B))
5		elxp3 3214	. . . . . 6 |- (z e. (A X. B) <-> E.xE.y(<.x, y>. = z /\ <.x, y>. e. (A X. B)))
6		snssi 2457	. . . . . . . . 9 |- (<.x, y>. e. (A X. B) -> {<.x, y>.} (_ (A X. B))
7		uniss 2511	. . . . . . . . . 10 |- ({<.x, y>.} (_ (A X. B) -> U.{<.x, y>.} (_ U.(A X. B))
8		opex 2772	. . . . . . . . . . 11 |- <.x, y>. e. V
9	8	unisn 2507	. . . . . . . . . 10 |- U.{<.x, y>.} = <.x, y>.
10	7, 9	syl5ssr 2096	. . . . . . . . 9 |- ({<.x, y>.} (_ (A X. B) -> <.x, y>. (_ U.(A X. B))
11		opnz 2785	. . . . . . . . . 10 |- -. <.x, y>. = (/)
12		sseq2 2073	. . . . . . . . . . . 12 |- (U.(A X. B) = (/) -> (<.x, y>. (_ U.(A X. B) <-> <.x, y>. (_ (/)))
13	12	biimpd 153	. . . . . . . . . . 11 |- (U.(A X. B) = (/) -> (<.x, y>. (_ U.(A X. B) -> <.x, y>. (_ (/)))
14		ss0 2293	. . . . . . . . . . 11 |- (<.x, y>. (_ (/) -> <.x, y>. = (/))
15	13, 14	syl6com 53	. . . . . . . . . 10 |- (<.x, y>. (_ U.(A X. B) -> (U.(A X. B) = (/) -> <.x, y>. = (/)))
16	11, 15	mtoi 107	. . . . . . . . 9 |- (<.x, y>. (_ U.(A X. B) -> -. U.(A X. B) = (/))
17	6, 10, 16	3syl 20	. . . . . . . 8 |- (<.x, y>. e. (A X. B) -> -. U.(A X. B) = (/))
18	17	adantl 388	. . . . . . 7 |- ((<.x, y>. = z /\ <.x, y>. e. (A X. B)) -> -. U.(A X. B) = (/))
19	18	19.23aivv 1291	. . . . . 6 |- (E.xE.y(<.x, y>. = z /\ <.x, y>. e. (A X. B)) -> -. U.(A X. B) = (/))
20	5, 19	sylbi 199	. . . . 5 |- (z e. (A X. B) -> -. U.(A X. B) = (/))
21	20	19.23aiv 1290	. . . 4 |- (E.z z e. (A X. B) -> -. U.(A X. B) = (/))
22	4, 21	sylbi 199	. . 3 |- (-. (A X. B) = (/) -> -. U.(A X. B) = (/))
23	22	a3i 74	. 2 |- (U.(A X. B) = (/) -> (A X. B) = (/))
24	3, 23	impbi 157	1 |- ((A X. B) = (/) <-> U.(A X. B) = (/))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   (_ wss 2037  (/)c0 2270  {csn 2399  <.cop 2401  U.cuni 2493   X. cxp 3158

This theorem is referenced by:  rankxpsuc 4687

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-opab 2657  df-xp 3174

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