Theorem xpexr 3465

Description: If a cross product is a set, one of its components must be a set.

Assertion

Ref	Expression
xpexr	|- ((A X. B) e. C -> (A e. V \/ B e. V))

Proof of Theorem xpexr

Step	Hyp	Ref	Expression
1		0ex 2701	. . . . . 6 |- (/) e. V
2		eleq1 1526	. . . . . 6 |- (A = (/) -> (A e. V <-> (/) e. V))
3	1, 2	mpbiri 194	. . . . 5 |- (A = (/) -> A e. V)
4	3	pm2.24d 105	. . . 4 |- (A = (/) -> (-. A e. V -> B e. V))
5	4	a1d 12	. . 3 |- (A = (/) -> ((A X. B) e. C -> (-. A e. V -> B e. V)))
6		rnxp 3458	. . . . . 6 |- (A =/= (/) -> ran ( A X. B) = B)
7	6	eleq1d 1532	. . . . 5 |- (A =/= (/) -> (ran ( A X. B) e. V <-> B e. V))
8		rnexg 3345	. . . . 5 |- ((A X. B) e. C -> ran ( A X. B) e. V)
9	7, 8	syl5bi 208	. . . 4 |- (A =/= (/) -> ((A X. B) e. C -> B e. V))
10	9	a1dd 42	. . 3 |- (A =/= (/) -> ((A X. B) e. C -> (-. A e. V -> B e. V)))
11	5, 10	pm2.61ine 1626	. 2 |- ((A X. B) e. C -> (-. A e. V -> B e. V))
12	11	orrd 233	1 |- ((A X. B) e. C -> (A e. V \/ B e. V))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802  (/)c0 2270   X. cxp 3158  ran crn 3161

This theorem is referenced by:  ismsg 7739

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-9 962  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-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

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-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-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179

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