Theorem xpexr2 4353

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 4177	. . . . . . 7 |- (B =/= (/) -> dom ( A X. B) = A)
2	1	adantl 424	. . . . . 6 |- (((A X. B) e. C /\ B =/= (/)) -> dom ( A X. B) = A)
3		dmexg 4206	. . . . . . 7 |- ((A X. B) e. C -> dom ( A X. B) e. _V)
4	3	adantr 425	. . . . . 6 |- (((A X. B) e. C /\ B =/= (/)) -> dom ( A X. B) e. _V)
5	2, 4	eqeltrrd 1972	. . . . 5 |- (((A X. B) e. C /\ B =/= (/)) -> A e. _V)
6		rnxp 4342	. . . . . . 7 |- (A =/= (/) -> ran ( A X. B) = B)
7	6	adantl 424	. . . . . 6 |- (((A X. B) e. C /\ A =/= (/)) -> ran ( A X. B) = B)
8		rnexg 4207	. . . . . . 7 |- ((A X. B) e. C -> ran ( A X. B) e. _V)
9	8	adantr 425	. . . . . 6 |- (((A X. B) e. C /\ A =/= (/)) -> ran ( A X. B) e. _V)
10	7, 9	eqeltrrd 1972	. . . . 5 |- (((A X. B) e. C /\ A =/= (/)) -> B e. _V)
11	5, 10	anim12i 360	. . . 4 |- ((((A X. B) e. C /\ B =/= (/)) /\ ((A X. B) e. C /\ A =/= (/))) -> (A e. _V /\ B e. _V))
12	11	anandis 570	. . 3 |- (((A X. B) e. C /\ (B =/= (/) /\ A =/= (/))) -> (A e. _V /\ B e. _V))
13	12	ancom2s 545	. 2 |- (((A X. B) e. C /\ (A =/= (/) /\ B =/= (/))) -> (A e. _V /\ B e. _V))
14		xpnz 4335	. 2 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
15	13, 14	sylan2br 502	1 |- (((A X. B) e. C /\ (A X. B) =/= (/)) -> (A e. _V /\ B e. _V))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875   X. cxp 3984  dom cdm 3986  ran crn 3987

This theorem is referenced by:  xpfi 5632

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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