Theorem cfslb 9088

Description: Any cofinal subset of A is at least as large as ( cf ` A ). (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfslb.1	|- A e. _V

Assertion

Ref	Expression
cfslb	|- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( cf ` A ) ~<_ B )

Proof of Theorem cfslb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . 3 |- ( card ` B ) e. _V
2		ssid 3624	. . . . . . 7 |- ( card ` B ) C_ ( card ` B )
3		cfslb.1	. . . . . . . . . . 11 |- A e. _V
4	3	ssex 4802	. . . . . . . . . 10 |- ( B C_ A -> B e. _V )
5	4	ad2antrr 762	. . . . . . . . 9 |- ( ( ( B C_ A /\ U. B = A ) /\ ( card ` B ) C_ ( card ` B ) ) -> B e. _V )
6		selpw 4165	. . . . . . . . . . . . 13 |- ( x e. ~P A <-> x C_ A )
7		sseq1 3626	. . . . . . . . . . . . 13 |- ( x = B -> ( x C_ A <-> B C_ A ) )
8	6, 7	syl5bb 272	. . . . . . . . . . . 12 |- ( x = B -> ( x e. ~P A <-> B C_ A ) )
9		unieq 4444	. . . . . . . . . . . . 13 |- ( x = B -> U. x = U. B )
10	9	eqeq1d 2624	. . . . . . . . . . . 12 |- ( x = B -> ( U. x = A <-> U. B = A ) )
11	8, 10	anbi12d 747	. . . . . . . . . . 11 |- ( x = B -> ( ( x e. ~P A /\ U. x = A ) <-> ( B C_ A /\ U. B = A ) ) )
12		fveq2 6191	. . . . . . . . . . . 12 |- ( x = B -> ( card ` x ) = ( card ` B ) )
13	12	sseq1d 3632	. . . . . . . . . . 11 |- ( x = B -> ( ( card ` x ) C_ ( card ` B ) <-> ( card ` B ) C_ ( card ` B ) ) )
14	11, 13	anbi12d 747	. . . . . . . . . 10 |- ( x = B -> ( ( ( x e. ~P A /\ U. x = A ) /\ ( card ` x ) C_ ( card ` B ) ) <-> ( ( B C_ A /\ U. B = A ) /\ ( card ` B ) C_ ( card ` B ) ) ) )
15	14	spcegv 3294	. . . . . . . . 9 |- ( B e. _V -> ( ( ( B C_ A /\ U. B = A ) /\ ( card ` B ) C_ ( card ` B ) ) -> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` x ) C_ ( card ` B ) ) ) )
16	5, 15	mpcom 38	. . . . . . . 8 |- ( ( ( B C_ A /\ U. B = A ) /\ ( card ` B ) C_ ( card ` B ) ) -> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` x ) C_ ( card ` B ) ) )
17		df-rex 2918	. . . . . . . . 9 |- ( E. x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) <-> E. x ( x e. { x e. ~P A | U. x = A } /\ ( card ` x ) C_ ( card ` B ) ) )
18		rabid 3116	. . . . . . . . . . 11 |- ( x e. { x e. ~P A | U. x = A } <-> ( x e. ~P A /\ U. x = A ) )
19	18	anbi1i 731	. . . . . . . . . 10 |- ( ( x e. { x e. ~P A | U. x = A } /\ ( card ` x ) C_ ( card ` B ) ) <-> ( ( x e. ~P A /\ U. x = A ) /\ ( card ` x ) C_ ( card ` B ) ) )
20	19	exbii 1774	. . . . . . . . 9 |- ( E. x ( x e. { x e. ~P A | U. x = A } /\ ( card ` x ) C_ ( card ` B ) ) <-> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` x ) C_ ( card ` B ) ) )
21	17, 20	bitri 264	. . . . . . . 8 |- ( E. x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) <-> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` x ) C_ ( card ` B ) ) )
22	16, 21	sylibr 224	. . . . . . 7 |- ( ( ( B C_ A /\ U. B = A ) /\ ( card ` B ) C_ ( card ` B ) ) -> E. x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) )
23	2, 22	mpan2 707	. . . . . 6 |- ( ( B C_ A /\ U. B = A ) -> E. x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) )
24		iinss 4571	. . . . . 6 |- ( E. x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) -> |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) )
25	23, 24	syl 17	. . . . 5 |- ( ( B C_ A /\ U. B = A ) -> |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) )
26	3	cflim3 9084	. . . . . 6 |- ( Lim A -> ( cf ` A ) = |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) )
27	26	sseq1d 3632	. . . . 5 |- ( Lim A -> ( ( cf ` A ) C_ ( card ` B ) <-> |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) C_ ( card ` B ) ) )
28	25, 27	syl5ibr 236	. . . 4 |- ( Lim A -> ( ( B C_ A /\ U. B = A ) -> ( cf ` A ) C_ ( card ` B ) ) )
29	28	3impib 1262	. . 3 |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( cf ` A ) C_ ( card ` B ) )
30		ssdomg 8001	. . 3 |- ( ( card ` B ) e. _V -> ( ( cf ` A ) C_ ( card ` B ) -> ( cf ` A ) ~<_ ( card ` B ) ) )
31	1, 29, 30	mpsyl 68	. 2 |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( cf ` A ) ~<_ ( card ` B ) )
32	4	adantl 482	. . . . 5 |- ( ( Lim A /\ B C_ A ) -> B e. _V )
33		limord 5784	. . . . . . 7 |- ( Lim A -> Ord A )
34		ordsson 6989	. . . . . . 7 |- ( Ord A -> A C_ On )
35	33, 34	syl 17	. . . . . 6 |- ( Lim A -> A C_ On )
36		sstr2 3610	. . . . . 6 |- ( B C_ A -> ( A C_ On -> B C_ On ) )
37	35, 36	mpan9 486	. . . . 5 |- ( ( Lim A /\ B C_ A ) -> B C_ On )
38		onssnum 8863	. . . . 5 |- ( ( B e. _V /\ B C_ On ) -> B e. dom card )
39	32, 37, 38	syl2anc 693	. . . 4 |- ( ( Lim A /\ B C_ A ) -> B e. dom card )
40		cardid2 8779	. . . 4 |- ( B e. dom card -> ( card ` B ) ~~ B )
41	39, 40	syl 17	. . 3 |- ( ( Lim A /\ B C_ A ) -> ( card ` B ) ~~ B )
42	41	3adant3 1081	. 2 |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( card ` B ) ~~ B )
43		domentr 8015	. 2 |- ( ( ( cf ` A ) ~<_ ( card ` B ) /\ ( card ` B ) ~~ B ) -> ( cf ` A ) ~<_ B )
44	31, 42, 43	syl2anc 693	1 |- ( ( Lim A /\ B C_ A /\ U. B = A ) -> ( cf ` A ) ~<_ B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|_ciin 4521   class class class wbr 4653   dom cdm 5114   Ord word 5722   Oncon0 5723   Lim wlim 5724   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   cardccrd 8761   cfccf 8763

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-rmo 2920  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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-card 8765  df-cf 8767

This theorem is referenced by:  cfslbn  9089  cfslb2n  9090  rankcf  9599