Theorem fin1a2lem9 9230

Description: Lemma for fin1a2 9237. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)

Assertion

Ref	Expression
fin1a2lem9	|- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> { b e. X | b ~<_ A } e. Fin )

Distinct variable groups:   A, b   X, b

Proof of Theorem fin1a2lem9

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

Step	Hyp	Ref	Expression
1		onfin2 8152	. . . . 5 |- om = ( On i^i Fin )
2		inss2 3834	. . . . 5 |- ( On i^i Fin ) C_ Fin
3	1, 2	eqsstri 3635	. . . 4 |- om C_ Fin
4		peano2 7086	. . . 4 |- ( A e. om -> suc A e. om )
5	3, 4	sseldi 3601	. . 3 |- ( A e. om -> suc A e. Fin )
6	5	3ad2ant3 1084	. 2 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> suc A e. Fin )
7	4	3ad2ant3 1084	. . 3 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> suc A e. om )
8		breq1 4656	. . . . . 6 |- ( b = c -> ( b ~<_ A <-> c ~<_ A ) )
9	8	elrab 3363	. . . . 5 |- ( c e. { b e. X | b ~<_ A } <-> ( c e. X /\ c ~<_ A ) )
10		simprr 796	. . . . . . . 8 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> c ~<_ A )
11		simpl2 1065	. . . . . . . . . . 11 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> X C_ Fin )
12		simprl 794	. . . . . . . . . . 11 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> c e. X )
13	11, 12	sseldd 3604	. . . . . . . . . 10 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> c e. Fin )
14		finnum 8774	. . . . . . . . . 10 |- ( c e. Fin -> c e. dom card )
15	13, 14	syl 17	. . . . . . . . 9 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> c e. dom card )
16		simpl3 1066	. . . . . . . . . . 11 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> A e. om )
17	3, 16	sseldi 3601	. . . . . . . . . 10 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> A e. Fin )
18		finnum 8774	. . . . . . . . . 10 |- ( A e. Fin -> A e. dom card )
19	17, 18	syl 17	. . . . . . . . 9 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> A e. dom card )
20		carddom2 8803	. . . . . . . . 9 |- ( ( c e. dom card /\ A e. dom card ) -> ( ( card ` c ) C_ ( card ` A ) <-> c ~<_ A ) )
21	15, 19, 20	syl2anc 693	. . . . . . . 8 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> ( ( card ` c ) C_ ( card ` A ) <-> c ~<_ A ) )
22	10, 21	mpbird 247	. . . . . . 7 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ c ~<_ A ) ) -> ( card ` c ) C_ ( card ` A ) )
23	22	ex 450	. . . . . 6 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( ( c e. X /\ c ~<_ A ) -> ( card ` c ) C_ ( card ` A ) ) )
24		cardnn 8789	. . . . . . . . 9 |- ( A e. om -> ( card ` A ) = A )
25	24	sseq2d 3633	. . . . . . . 8 |- ( A e. om -> ( ( card ` c ) C_ ( card ` A ) <-> ( card ` c ) C_ A ) )
26		cardon 8770	. . . . . . . . 9 |- ( card ` c ) e. On
27		nnon 7071	. . . . . . . . 9 |- ( A e. om -> A e. On )
28		onsssuc 5813	. . . . . . . . 9 |- ( ( ( card ` c ) e. On /\ A e. On ) -> ( ( card ` c ) C_ A <-> ( card ` c ) e. suc A ) )
29	26, 27, 28	sylancr 695	. . . . . . . 8 |- ( A e. om -> ( ( card ` c ) C_ A <-> ( card ` c ) e. suc A ) )
30	25, 29	bitrd 268	. . . . . . 7 |- ( A e. om -> ( ( card ` c ) C_ ( card ` A ) <-> ( card ` c ) e. suc A ) )
31	30	3ad2ant3 1084	. . . . . 6 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( ( card ` c ) C_ ( card ` A ) <-> ( card ` c ) e. suc A ) )
32	23, 31	sylibd 229	. . . . 5 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( ( c e. X /\ c ~<_ A ) -> ( card ` c ) e. suc A ) )
33	9, 32	syl5bi 232	. . . 4 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( c e. { b e. X | b ~<_ A } -> ( card ` c ) e. suc A ) )
34		elrabi 3359	. . . . 5 |- ( c e. { b e. X | b ~<_ A } -> c e. X )
35		elrabi 3359	. . . . 5 |- ( d e. { b e. X | b ~<_ A } -> d e. X )
36		ssel 3597	. . . . . . . . . . 11 |- ( X C_ Fin -> ( c e. X -> c e. Fin ) )
37		ssel 3597	. . . . . . . . . . 11 |- ( X C_ Fin -> ( d e. X -> d e. Fin ) )
38	36, 37	anim12d 586	. . . . . . . . . 10 |- ( X C_ Fin -> ( ( c e. X /\ d e. X ) -> ( c e. Fin /\ d e. Fin ) ) )
39	38	imp 445	. . . . . . . . 9 |- ( ( X C_ Fin /\ ( c e. X /\ d e. X ) ) -> ( c e. Fin /\ d e. Fin ) )
40	39	3ad2antl2 1224	. . . . . . . 8 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ d e. X ) ) -> ( c e. Fin /\ d e. Fin ) )
41		sorpssi 6943	. . . . . . . . 9 |- ( ( [ C.] Or X /\ ( c e. X /\ d e. X ) ) -> ( c C_ d \/ d C_ c ) )
42	41	3ad2antl1 1223	. . . . . . . 8 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ d e. X ) ) -> ( c C_ d \/ d C_ c ) )
43		finnum 8774	. . . . . . . . . . 11 |- ( d e. Fin -> d e. dom card )
44		carden2 8813	. . . . . . . . . . 11 |- ( ( c e. dom card /\ d e. dom card ) -> ( ( card ` c ) = ( card ` d ) <-> c ~~ d ) )
45	14, 43, 44	syl2an 494	. . . . . . . . . 10 |- ( ( c e. Fin /\ d e. Fin ) -> ( ( card ` c ) = ( card ` d ) <-> c ~~ d ) )
46	45	adantr 481	. . . . . . . . 9 |- ( ( ( c e. Fin /\ d e. Fin ) /\ ( c C_ d \/ d C_ c ) ) -> ( ( card ` c ) = ( card ` d ) <-> c ~~ d ) )
47		fin23lem25 9146	. . . . . . . . . . 11 |- ( ( c e. Fin /\ d e. Fin /\ ( c C_ d \/ d C_ c ) ) -> ( c ~~ d <-> c = d ) )
48	47	3expa 1265	. . . . . . . . . 10 |- ( ( ( c e. Fin /\ d e. Fin ) /\ ( c C_ d \/ d C_ c ) ) -> ( c ~~ d <-> c = d ) )
49	48	biimpd 219	. . . . . . . . 9 |- ( ( ( c e. Fin /\ d e. Fin ) /\ ( c C_ d \/ d C_ c ) ) -> ( c ~~ d -> c = d ) )
50	46, 49	sylbid 230	. . . . . . . 8 |- ( ( ( c e. Fin /\ d e. Fin ) /\ ( c C_ d \/ d C_ c ) ) -> ( ( card ` c ) = ( card ` d ) -> c = d ) )
51	40, 42, 50	syl2anc 693	. . . . . . 7 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ d e. X ) ) -> ( ( card ` c ) = ( card ` d ) -> c = d ) )
52		fveq2 6191	. . . . . . 7 |- ( c = d -> ( card ` c ) = ( card ` d ) )
53	51, 52	impbid1 215	. . . . . 6 |- ( ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) /\ ( c e. X /\ d e. X ) ) -> ( ( card ` c ) = ( card ` d ) <-> c = d ) )
54	53	ex 450	. . . . 5 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( ( c e. X /\ d e. X ) -> ( ( card ` c ) = ( card ` d ) <-> c = d ) ) )
55	34, 35, 54	syl2ani 688	. . . 4 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( ( c e. { b e. X | b ~<_ A } /\ d e. { b e. X | b ~<_ A } ) -> ( ( card ` c ) = ( card ` d ) <-> c = d ) ) )
56	33, 55	dom2d 7996	. . 3 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> ( suc A e. om -> { b e. X | b ~<_ A } ~<_ suc A ) )
57	7, 56	mpd 15	. 2 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> { b e. X | b ~<_ A } ~<_ suc A )
58		domfi 8181	. 2 |- ( ( suc A e. Fin /\ { b e. X | b ~<_ A } ~<_ suc A ) -> { b e. X | b ~<_ A } e. Fin )
59	6, 57, 58	syl2anc 693	1 |- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> { b e. X | b ~<_ A } e. Fin )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   i^i cin 3573   C_ wss 3574   class class class wbr 4653   Or wor 5034   dom cdm 5114   Oncon0 5723   suc csuc 5725   ` cfv 5888   [ C.] crpss 6936   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   cardccrd 8761

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-rep 4771  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-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-rpss 6937  df-om 7066  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin1a2lem11  9232