Theorem unixp0 3510

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 2505	. . 3 |- ((A X. B) = (/) -> U.(A X. B) = U.(/))
2		uni0 2520	. . 3 |- U.(/) = (/)
3	1, 2	syl6eq 1520	. 2 |- ((A X. B) = (/) -> U.(A X. B) = (/))
4		n0 2285	. . . 4 |- (-. (A X. B) = (/) <-> E.z z e. (A X. B))
5		elxp3 3219	. . . . . 6 |- (z e. (A X. B) <-> E.xE.y(<.x, y>. = z /\ <.x, y>. e. (A X. B)))
6		snssi 2462	. . . . . . . . 9 |- (<.x, y>. e. (A X. B) -> {<.x, y>.} (_ (A X. B))
7		uniss 2516	. . . . . . . . . 10 |- ({<.x, y>.} (_ (A X. B) -> U.{<.x, y>.} (_ U.(A X. B))
8		opex 2777	. . . . . . . . . . 11 |- <.x, y>. e. V
9	8	unisn 2512	. . . . . . . . . 10 |- U.{<.x, y>.} = <.x, y>.
10	7, 9	syl5ssr 2102	. . . . . . . . 9 |- ({<.x, y>.} (_ (A X. B) -> <.x, y>. (_ U.(A X. B))
11		opnz 2790	. . . . . . . . . 10 |- -. <.x, y>. = (/)
12		sseq2 2079	. . . . . . . . . . . 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 2299	. . . . . . . . . . 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 1294	. . . . . 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 1293	. . . 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 954   e. wcel 956  E.wex 978   (_ wss 2043  (/)c0 2276  {csn 2405  <.cop 2407  U.cuni 2498   X. cxp 3163

This theorem is referenced by:  rankxpsuc 4695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-opab 2662  df-xp 3179

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