Theorem xpexr 7106

Description: If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)

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 4790	. . . . . 6 |- (/) e. _V
2		eleq1 2689	. . . . . 6 |- ( A = (/) -> ( A e. _V <-> (/) e. _V ) )
3	1, 2	mpbiri 248	. . . . 5 |- ( A = (/) -> A e. _V )
4	3	pm2.24d 147	. . . 4 |- ( A = (/) -> ( -. A e. _V -> B e. _V ) )
5	4	a1d 25	. . 3 |- ( A = (/) -> ( ( A X. B ) e. C -> ( -. A e. _V -> B e. _V ) ) )
6		rnexg 7098	. . . . 5 |- ( ( A X. B ) e. C -> ran ( A X. B ) e. _V )
7		rnxp 5564	. . . . . 6 |- ( A =/= (/) -> ran ( A X. B ) = B )
8	7	eleq1d 2686	. . . . 5 |- ( A =/= (/) -> ( ran ( A X. B ) e. _V <-> B e. _V ) )
9	6, 8	syl5ib 234	. . . 4 |- ( A =/= (/) -> ( ( A X. B ) e. C -> B e. _V ) )
10	9	a1dd 50	. . 3 |- ( A =/= (/) -> ( ( A X. B ) e. C -> ( -. A e. _V -> B e. _V ) ) )
11	5, 10	pm2.61ine 2877	. 2 |- ( ( A X. B ) e. C -> ( -. A e. _V -> B e. _V ) )
12	11	orrd 393	1 |- ( ( A X. B ) e. C -> ( A e. _V \/ B e. _V ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   X. cxp 5112   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)