Theorem cfss 9087

Description: There is a cofinal subset of A of cardinality ( cf ` A ). (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfss.1	|- A e. _V

Assertion

Ref	Expression
cfss	|- ( Lim A -> E. x ( x C_ A /\ x ~~ ( cf ` A ) /\ U. x = A ) )

Distinct variable group:   x, A

Proof of Theorem cfss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfss.1	. . . . . 6 |- A e. _V
2	1	cflim3 9084	. . . . 5 |- ( Lim A -> ( cf ` A ) = |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) )
3		fvex 6201	. . . . . . 7 |- ( card ` x ) e. _V
4	3	dfiin2 4555	. . . . . 6 |- |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) = |^| { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) }
5		cardon 8770	. . . . . . . . . 10 |- ( card ` x ) e. On
6		eleq1 2689	. . . . . . . . . 10 |- ( y = ( card ` x ) -> ( y e. On <-> ( card ` x ) e. On ) )
7	5, 6	mpbiri 248	. . . . . . . . 9 |- ( y = ( card ` x ) -> y e. On )
8	7	rexlimivw 3029	. . . . . . . 8 |- ( E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) -> y e. On )
9	8	abssi 3677	. . . . . . 7 |- { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } C_ On
10		limuni 5785	. . . . . . . . . . . 12 |- ( Lim A -> A = U. A )
11	10	eqcomd 2628	. . . . . . . . . . 11 |- ( Lim A -> U. A = A )
12		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( x = A -> ( card ` x ) = ( card ` A ) )
13	12	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( x = A -> ( card ` A ) = ( card ` x ) )
14	13	biantrud 528	. . . . . . . . . . . . 13 |- ( x = A -> ( U. A = A <-> ( U. A = A /\ ( card ` A ) = ( card ` x ) ) ) )
15		unieq 4444	. . . . . . . . . . . . . . . 16 |- ( x = A -> U. x = U. A )
16	15	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( x = A -> ( U. x = A <-> U. A = A ) )
17	1	pwid 4174	. . . . . . . . . . . . . . . . 17 |- A e. ~P A
18		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( x = A -> ( x e. ~P A <-> A e. ~P A ) )
19	17, 18	mpbiri 248	. . . . . . . . . . . . . . . 16 |- ( x = A -> x e. ~P A )
20	19	biantrurd 529	. . . . . . . . . . . . . . 15 |- ( x = A -> ( U. x = A <-> ( x e. ~P A /\ U. x = A ) ) )
21	16, 20	bitr3d 270	. . . . . . . . . . . . . 14 |- ( x = A -> ( U. A = A <-> ( x e. ~P A /\ U. x = A ) ) )
22	21	anbi1d 741	. . . . . . . . . . . . 13 |- ( x = A -> ( ( U. A = A /\ ( card ` A ) = ( card ` x ) ) <-> ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) ) )
23	14, 22	bitr2d 269	. . . . . . . . . . . 12 |- ( x = A -> ( ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) <-> U. A = A ) )
24	1, 23	spcev 3300	. . . . . . . . . . 11 |- ( U. A = A -> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) )
25	11, 24	syl 17	. . . . . . . . . 10 |- ( Lim A -> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) )
26		df-rex 2918	. . . . . . . . . . 11 |- ( E. x e. { x e. ~P A | U. x = A } ( card ` A ) = ( card ` x ) <-> E. x ( x e. { x e. ~P A | U. x = A } /\ ( card ` A ) = ( card ` x ) ) )
27		rabid 3116	. . . . . . . . . . . . 13 |- ( x e. { x e. ~P A | U. x = A } <-> ( x e. ~P A /\ U. x = A ) )
28	27	anbi1i 731	. . . . . . . . . . . 12 |- ( ( x e. { x e. ~P A | U. x = A } /\ ( card ` A ) = ( card ` x ) ) <-> ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) )
29	28	exbii 1774	. . . . . . . . . . 11 |- ( E. x ( x e. { x e. ~P A | U. x = A } /\ ( card ` A ) = ( card ` x ) ) <-> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) )
30	26, 29	bitri 264	. . . . . . . . . 10 |- ( E. x e. { x e. ~P A | U. x = A } ( card ` A ) = ( card ` x ) <-> E. x ( ( x e. ~P A /\ U. x = A ) /\ ( card ` A ) = ( card ` x ) ) )
31	25, 30	sylibr 224	. . . . . . . . 9 |- ( Lim A -> E. x e. { x e. ~P A | U. x = A } ( card ` A ) = ( card ` x ) )
32		fvex 6201	. . . . . . . . . 10 |- ( card ` A ) e. _V
33		eqeq1 2626	. . . . . . . . . . 11 |- ( y = ( card ` A ) -> ( y = ( card ` x ) <-> ( card ` A ) = ( card ` x ) ) )
34	33	rexbidv 3052	. . . . . . . . . 10 |- ( y = ( card ` A ) -> ( E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) <-> E. x e. { x e. ~P A | U. x = A } ( card ` A ) = ( card ` x ) ) )
35	32, 34	spcev 3300	. . . . . . . . 9 |- ( E. x e. { x e. ~P A | U. x = A } ( card ` A ) = ( card ` x ) -> E. y E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) )
36	31, 35	syl 17	. . . . . . . 8 |- ( Lim A -> E. y E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) )
37		abn0 3954	. . . . . . . 8 |- ( { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } =/= (/) <-> E. y E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) )
38	36, 37	sylibr 224	. . . . . . 7 |- ( Lim A -> { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } =/= (/) )
39		onint 6995	. . . . . . 7 |- ( ( { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } C_ On /\ { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } =/= (/) ) -> |^| { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } e. { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } )
40	9, 38, 39	sylancr 695	. . . . . 6 |- ( Lim A -> |^| { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } e. { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } )
41	4, 40	syl5eqel 2705	. . . . 5 |- ( Lim A -> |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) e. { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } )
42	2, 41	eqeltrd 2701	. . . 4 |- ( Lim A -> ( cf ` A ) e. { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } )
43		fvex 6201	. . . . 5 |- ( cf ` A ) e. _V
44		eqeq1 2626	. . . . . 6 |- ( y = ( cf ` A ) -> ( y = ( card ` x ) <-> ( cf ` A ) = ( card ` x ) ) )
45	44	rexbidv 3052	. . . . 5 |- ( y = ( cf ` A ) -> ( E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) <-> E. x e. { x e. ~P A | U. x = A } ( cf ` A ) = ( card ` x ) ) )
46	43, 45	elab 3350	. . . 4 |- ( ( cf ` A ) e. { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } <-> E. x e. { x e. ~P A | U. x = A } ( cf ` A ) = ( card ` x ) )
47	42, 46	sylib 208	. . 3 |- ( Lim A -> E. x e. { x e. ~P A | U. x = A } ( cf ` A ) = ( card ` x ) )
48		df-rex 2918	. . 3 |- ( E. x e. { x e. ~P A | U. x = A } ( cf ` A ) = ( card ` x ) <-> E. x ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) )
49	47, 48	sylib 208	. 2 |- ( Lim A -> E. x ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) )
50		simprl 794	. . . . . . . 8 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> x e. { x e. ~P A | U. x = A } )
51	50, 27	sylib 208	. . . . . . 7 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> ( x e. ~P A /\ U. x = A ) )
52	51	simpld 475	. . . . . 6 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> x e. ~P A )
53	52	elpwid 4170	. . . . 5 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> x C_ A )
54		simpl 473	. . . . . . 7 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> Lim A )
55		vex 3203	. . . . . . . . . 10 |- x e. _V
56		limord 5784	. . . . . . . . . . . 12 |- ( Lim A -> Ord A )
57		ordsson 6989	. . . . . . . . . . . 12 |- ( Ord A -> A C_ On )
58	56, 57	syl 17	. . . . . . . . . . 11 |- ( Lim A -> A C_ On )
59		sstr 3611	. . . . . . . . . . 11 |- ( ( x C_ A /\ A C_ On ) -> x C_ On )
60	58, 59	sylan2 491	. . . . . . . . . 10 |- ( ( x C_ A /\ Lim A ) -> x C_ On )
61		onssnum 8863	. . . . . . . . . 10 |- ( ( x e. _V /\ x C_ On ) -> x e. dom card )
62	55, 60, 61	sylancr 695	. . . . . . . . 9 |- ( ( x C_ A /\ Lim A ) -> x e. dom card )
63		cardid2 8779	. . . . . . . . 9 |- ( x e. dom card -> ( card ` x ) ~~ x )
64	62, 63	syl 17	. . . . . . . 8 |- ( ( x C_ A /\ Lim A ) -> ( card ` x ) ~~ x )
65	64	ensymd 8007	. . . . . . 7 |- ( ( x C_ A /\ Lim A ) -> x ~~ ( card ` x ) )
66	53, 54, 65	syl2anc 693	. . . . . 6 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> x ~~ ( card ` x ) )
67		simprr 796	. . . . . 6 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> ( cf ` A ) = ( card ` x ) )
68	66, 67	breqtrrd 4681	. . . . 5 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> x ~~ ( cf ` A ) )
69	51	simprd 479	. . . . 5 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> U. x = A )
70	53, 68, 69	3jca 1242	. . . 4 |- ( ( Lim A /\ ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) ) -> ( x C_ A /\ x ~~ ( cf ` A ) /\ U. x = A ) )
71	70	ex 450	. . 3 |- ( Lim A -> ( ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) -> ( x C_ A /\ x ~~ ( cf ` A ) /\ U. x = A ) ) )
72	71	eximdv 1846	. 2 |- ( Lim A -> ( E. x ( x e. { x e. ~P A | U. x = A } /\ ( cf ` A ) = ( card ` x ) ) -> E. x ( x C_ A /\ x ~~ ( cf ` A ) /\ U. x = A ) ) )
73	49, 72	mpd 15	1 |- ( Lim A -> E. x ( x C_ A /\ x ~~ ( cf ` A ) /\ U. x = A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   |^|_ciin 4521   class class class wbr 4653   dom cdm 5114   Ord word 5722   Oncon0 5723   Lim wlim 5724   ` cfv 5888   ~~ cen 7952   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: (None)