Theorem xpfir 8182

Description: The components of a nonempty finite Cartesian product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
xpfir	|- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> ( A e. Fin /\ B e. Fin ) )

Proof of Theorem xpfir

Step	Hyp	Ref	Expression
1		xpexr2 7107	. . . . 5 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> ( A e. _V /\ B e. _V ) )
2	1	simpld 475	. . . 4 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> A e. _V )
3	1	simprd 479	. . . 4 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> B e. _V )
4		simpr 477	. . . . . 6 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> ( A X. B ) =/= (/) )
5		xpnz 5553	. . . . . 6 |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )
6	4, 5	sylibr 224	. . . . 5 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> ( A =/= (/) /\ B =/= (/) ) )
7	6	simprd 479	. . . 4 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> B =/= (/) )
8		xpdom3 8058	. . . 4 |- ( ( A e. _V /\ B e. _V /\ B =/= (/) ) -> A ~<_ ( A X. B ) )
9	2, 3, 7, 8	syl3anc 1326	. . 3 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> A ~<_ ( A X. B ) )
10		domfi 8181	. . 3 |- ( ( ( A X. B ) e. Fin /\ A ~<_ ( A X. B ) ) -> A e. Fin )
11	9, 10	syldan 487	. 2 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> A e. Fin )
12	6	simpld 475	. . . . 5 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> A =/= (/) )
13		xpdom3 8058	. . . . 5 |- ( ( B e. _V /\ A e. _V /\ A =/= (/) ) -> B ~<_ ( B X. A ) )
14	3, 2, 12, 13	syl3anc 1326	. . . 4 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> B ~<_ ( B X. A ) )
15		xpcomeng 8052	. . . . 5 |- ( ( B e. _V /\ A e. _V ) -> ( B X. A ) ~~ ( A X. B ) )
16	3, 2, 15	syl2anc 693	. . . 4 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> ( B X. A ) ~~ ( A X. B ) )
17		domentr 8015	. . . 4 |- ( ( B ~<_ ( B X. A ) /\ ( B X. A ) ~~ ( A X. B ) ) -> B ~<_ ( A X. B ) )
18	14, 16, 17	syl2anc 693	. . 3 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> B ~<_ ( A X. B ) )
19		domfi 8181	. . 3 |- ( ( ( A X. B ) e. Fin /\ B ~<_ ( A X. B ) ) -> B e. Fin )
20	18, 19	syldan 487	. 2 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> B e. Fin )
21	11, 20	jca 554	1 |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) -> ( A e. Fin /\ B e. Fin ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   X. cxp 5112   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955

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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959

This theorem is referenced by: (None)