Theorem xpexr2 3472

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 3327	. . . . . . 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 3352	. . . . . . 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 1546	. . . . 5 |- (((A X. B) e. C /\ B =/= (/)) -> A e. V)
6		rnxp 3464	. . . . . . 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 3353	. . . . . . 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 1546	. . . . 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 3458	. 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 954   e. wcel 956   =/= wne 1582  Vcvv 1807  (/)c0 2276   X. cxp 3163  dom cdm 3165  ran crn 3166

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-9 963  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  ax-un 2861

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-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-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184

Copyright terms: Public domain