Theorem fin12 9235

Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 9237. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
fin12	|- ( A e. Fin -> A e. FinII )

Proof of Theorem fin12

Dummy variables b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . 8 |- b e. _V
2	1	a1i 11	. . . . . . 7 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> b e. _V )
3		isfin1-3 9208	. . . . . . . . 9 |- ( A e. Fin -> ( A e. Fin <-> `' [ C.] Fr ~P A ) )
4	3	ibi 256	. . . . . . . 8 |- ( A e. Fin -> `' [ C.] Fr ~P A )
5	4	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> `' [ C.] Fr ~P A )
6		elpwi 4168	. . . . . . . 8 |- ( b e. ~P ~P A -> b C_ ~P A )
7	6	ad2antlr 763	. . . . . . 7 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> b C_ ~P A )
8		simprl 794	. . . . . . 7 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> b =/= (/) )
9		fri 5076	. . . . . . 7 |- ( ( ( b e. _V /\ `' [ C.] Fr ~P A ) /\ ( b C_ ~P A /\ b =/= (/) ) ) -> E. c e. b A. d e. b -. d `' [ C.] c )
10	2, 5, 7, 8, 9	syl22anc 1327	. . . . . 6 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> E. c e. b A. d e. b -. d `' [ C.] c )
11		vex 3203	. . . . . . . . . . 11 |- d e. _V
12		vex 3203	. . . . . . . . . . 11 |- c e. _V
13	11, 12	brcnv 5305	. . . . . . . . . 10 |- ( d `' [ C.] c <-> c [ C.] d )
14	11	brrpss 6940	. . . . . . . . . 10 |- ( c [ C.] d <-> c C. d )
15	13, 14	bitri 264	. . . . . . . . 9 |- ( d `' [ C.] c <-> c C. d )
16	15	notbii 310	. . . . . . . 8 |- ( -. d `' [ C.] c <-> -. c C. d )
17	16	ralbii 2980	. . . . . . 7 |- ( A. d e. b -. d `' [ C.] c <-> A. d e. b -. c C. d )
18	17	rexbii 3041	. . . . . 6 |- ( E. c e. b A. d e. b -. d `' [ C.] c <-> E. c e. b A. d e. b -. c C. d )
19	10, 18	sylib 208	. . . . 5 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> E. c e. b A. d e. b -. c C. d )
20		sorpssuni 6946	. . . . . 6 |- ( [ C.] Or b -> ( E. c e. b A. d e. b -. c C. d <-> U. b e. b ) )
21	20	ad2antll 765	. . . . 5 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> ( E. c e. b A. d e. b -. c C. d <-> U. b e. b ) )
22	19, 21	mpbid 222	. . . 4 |- ( ( ( A e. Fin /\ b e. ~P ~P A ) /\ ( b =/= (/) /\ [ C.] Or b ) ) -> U. b e. b )
23	22	ex 450	. . 3 |- ( ( A e. Fin /\ b e. ~P ~P A ) -> ( ( b =/= (/) /\ [ C.] Or b ) -> U. b e. b ) )
24	23	ralrimiva 2966	. 2 |- ( A e. Fin -> A. b e. ~P ~P A ( ( b =/= (/) /\ [ C.] Or b ) -> U. b e. b ) )
25		isfin2 9116	. 2 |- ( A e. Fin -> ( A e. FinII <-> A. b e. ~P ~P A ( ( b =/= (/) /\ [ C.] Or b ) -> U. b e. b ) ) )
26	24, 25	mpbird 247	1 |- ( A e. Fin -> A e. FinII )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   C. wpss 3575   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   Or wor 5034   Fr wfr 5070   `'ccnv 5113   [ C.] crpss 6936   Fincfn 7955  Fin^IIcfin2 9101

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-pred 5680  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fin2 9108

This theorem is referenced by:  fin1a2s  9236  fin1a2  9237  finngch  9477