Theorem xpexr2 3466

Description: If a nonempty cross product is a set, so are both of its components.

Assertion

Ref	Expression
xpexr2	|- (((A X. B) e. C /\ (A X. B) =/= (/)) -> (A e. V /\ B e. V))

Proof of Theorem xpexr2

Step	Hyp	Ref	Expression
1		dmxp 3321	. . . . . . 7 |- (B =/= (/) -> dom ( A X. B) = A)
2	1	adantl 388	. . . . . 6 |- (((A X. B) e. C /\ B =/= (/)) -> dom ( A X. B) = A)
3		dmexg 3344	. . . . . . 7 |- ((A X. B) e. C -> dom ( A X. B) e. V)
4	3	adantr 389	. . . . . 6 |- (((A X. B) e. C /\ B =/= (/)) -> dom ( A X. B) e. V)
5	2, 4	eqeltrrd 1541	. . . . 5 |- (((A X. B) e. C /\ B =/= (/)) -> A e. V)
6		rnxp 3458	. . . . . . 7 |- (A =/= (/) -> ran ( A X. B) = B)
7	6	adantl 388	. . . . . 6 |- (((A X. B) e. C /\ A =/= (/)) -> ran ( A X. B) = B)
8		rnexg 3345	. . . . . . 7 |- ((A X. B) e. C -> ran ( A X. B) e. V)
9	8	adantr 389	. . . . . 6 |- (((A X. B) e. C /\ A =/= (/)) -> ran ( A X. B) e. V)
10	7, 9	eqeltrrd 1541	. . . . 5 |- (((A X. B) e. C /\ A =/= (/)) -> B e. V)
11	5, 10	anim12i 333	. . . 4 |- ((((A X. B) e. C /\ B =/= (/)) /\ ((A X. B) e. C /\ A =/= (/))) -> (A e. V /\ B e. V))
12	11	anandis 511	. . 3 |- (((A X. B) e. C /\ (B =/= (/) /\ A =/= (/))) -> (A e. V /\ B e. V))
13	12	ancom2s 486	. 2 |- (((A X. B) e. C /\ (A =/= (/) /\ B =/= (/))) -> (A e. V /\ B e. V))
14		xpnz 3452	. 2 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
15	13, 14	sylan2br 453	1 |- (((A X. B) e. C /\ (A X. B) =/= (/)) -> (A e. V /\ B e. V))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802  (/)c0 2270   X. cxp 3158  dom cdm 3160  ran crn 3161

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-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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