Theorem xpexr2 5111

Description: If a nonempty cross product is a set, so are both of its components. (Contributed by set.mm contributors, 27-Aug-2006.) (Revised by set.mm contributors, 5-May-2007.)

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		xpnz 5046	. 2 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2		dmxp 4924	. . . . . 6 |- (B =/= (/) -> dom (A X. B) = A)
3	2	adantl 452	. . . . 5 |- (((A X. B) e. C /\ B =/= (/)) -> dom (A X. B) = A)
4		dmexg 5106	. . . . . 6 |- ((A X. B) e. C -> dom (A X. B) e. _V)
5	4	adantr 451	. . . . 5 |- (((A X. B) e. C /\ B =/= (/)) -> dom (A X. B) e. _V)
6	3, 5	eqeltrrd 2428	. . . 4 |- (((A X. B) e. C /\ B =/= (/)) -> A e. _V)
7		rnxp 5052	. . . . . 6 |- (A =/= (/) -> ran (A X. B) = B)
8	7	adantl 452	. . . . 5 |- (((A X. B) e. C /\ A =/= (/)) -> ran (A X. B) = B)
9		rnexg 5105	. . . . . 6 |- ((A X. B) e. C -> ran (A X. B) e. _V)
10	9	adantr 451	. . . . 5 |- (((A X. B) e. C /\ A =/= (/)) -> ran (A X. B) e. _V)
11	8, 10	eqeltrrd 2428	. . . 4 |- (((A X. B) e. C /\ A =/= (/)) -> B e. _V)
12	6, 11	anim12dan 810	. . 3 |- (((A X. B) e. C /\ (B =/= (/) /\ A =/= (/))) -> (A e. _V /\ B e. _V))
13	12	ancom2s 777	. 2 |- (((A X. B) e. C /\ (A =/= (/) /\ B =/= (/))) -> (A e. _V /\ B e. _V))
14	1, 13	sylan2br 462	1 |- (((A X. B) e. C /\ (A X. B) =/= (/)) -> (A e. _V /\ B e. _V))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860  (/)c0 3551   X. cxp 4771  dom cdm 4773  ran crn 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by: (None)