Theorem xpexr2 3477

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 3329	. . . . . . 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 3355	. . . . . . 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 1548	. . . . 5 |- (((A X. B) e. C /\ B =/= (/)) -> A e. V)
6		rnxp 3469	. . . . . . 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 3356	. . . . . . 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 1548	. . . . 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 512	. . 3 |- (((A X. B) e. C /\ (B =/= (/) /\ A =/= (/))) -> (A e. V /\ B e. V))
13	12	ancom2s 487	. 2 |- (((A X. B) e. C /\ (A =/= (/) /\ B =/= (/))) -> (A e. V /\ B e. V))
14		xpnz 3463	. 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 955   e. wcel 957   =/= wne 1584  Vcvv 1809  (/)c0 2278   X. cxp 3165  dom cdm 3167  ran crn 3168

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-xp 3181  df-rel 3182  df-cnv 3183  df-dm 3185  df-rn 3186

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