Theorem winalim2 9518

Description: A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)

Assertion

Ref	Expression
winalim2	|- ( ( A e. InaccW /\ A =/= om ) -> E. x ( ( aleph ` x ) = A /\ Lim x ) )

Distinct variable group:   x, A

Proof of Theorem winalim2

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		winacard 9514	. . . 4 |- ( A e. InaccW -> ( card ` A ) = A )
2		winainf 9516	. . . . 5 |- ( A e. InaccW -> om C_ A )
3		cardalephex 8913	. . . . 5 |- ( om C_ A -> ( ( card ` A ) = A <-> E. x e. On A = ( aleph ` x ) ) )
4	2, 3	syl 17	. . . 4 |- ( A e. InaccW -> ( ( card ` A ) = A <-> E. x e. On A = ( aleph ` x ) ) )
5	1, 4	mpbid 222	. . 3 |- ( A e. InaccW -> E. x e. On A = ( aleph ` x ) )
6	5	adantr 481	. 2 |- ( ( A e. InaccW /\ A =/= om ) -> E. x e. On A = ( aleph ` x ) )
7		df-rex 2918	. . 3 |- ( E. x e. On A = ( aleph ` x ) <-> E. x ( x e. On /\ A = ( aleph ` x ) ) )
8		simprr 796	. . . . . . 7 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> A = ( aleph ` x ) )
9	8	eqcomd 2628	. . . . . 6 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( aleph ` x ) = A )
10		simprl 794	. . . . . . . 8 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> x e. On )
11		onzsl 7046	. . . . . . . 8 |- ( x e. On <-> ( x = (/) \/ E. y e. On x = suc y \/ ( x e. _V /\ Lim x ) ) )
12	10, 11	sylib 208	. . . . . . 7 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( x = (/) \/ E. y e. On x = suc y \/ ( x e. _V /\ Lim x ) ) )
13		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> A =/= om )
14		fveq2 6191	. . . . . . . . . . . . . 14 |- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) )
15		aleph0 8889	. . . . . . . . . . . . . 14 |- ( aleph ` (/) ) = om
16	14, 15	syl6eq 2672	. . . . . . . . . . . . 13 |- ( x = (/) -> ( aleph ` x ) = om )
17		eqtr 2641	. . . . . . . . . . . . 13 |- ( ( A = ( aleph ` x ) /\ ( aleph ` x ) = om ) -> A = om )
18	16, 17	sylan2 491	. . . . . . . . . . . 12 |- ( ( A = ( aleph ` x ) /\ x = (/) ) -> A = om )
19	18	ex 450	. . . . . . . . . . 11 |- ( A = ( aleph ` x ) -> ( x = (/) -> A = om ) )
20	19	necon3ad 2807	. . . . . . . . . 10 |- ( A = ( aleph ` x ) -> ( A =/= om -> -. x = (/) ) )
21	8, 13, 20	sylc 65	. . . . . . . . 9 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> -. x = (/) )
22	21	pm2.21d 118	. . . . . . . 8 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( x = (/) -> Lim x ) )
23		suceloni 7013	. . . . . . . . . . . . . . . 16 |- ( y e. On -> suc y e. On )
24		vex 3203	. . . . . . . . . . . . . . . . 17 |- y e. _V
25	24	sucid 5804	. . . . . . . . . . . . . . . 16 |- y e. suc y
26		alephord2i 8900	. . . . . . . . . . . . . . . 16 |- ( suc y e. On -> ( y e. suc y -> ( aleph ` y ) e. ( aleph ` suc y ) ) )
27	23, 25, 26	mpisyl 21	. . . . . . . . . . . . . . 15 |- ( y e. On -> ( aleph ` y ) e. ( aleph ` suc y ) )
28	27	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( aleph ` y ) e. ( aleph ` suc y ) )
29		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> A = ( aleph ` x ) )
30		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) )
31	30	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( aleph ` x ) = ( aleph ` suc y ) )
32	29, 31	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> A = ( aleph ` suc y ) )
33	28, 32	eleqtrrd 2704	. . . . . . . . . . . . 13 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( aleph ` y ) e. A )
34		elwina 9508	. . . . . . . . . . . . . . 15 |- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. z e. A E. w e. A z ~< w ) )
35	34	simp3bi 1078	. . . . . . . . . . . . . 14 |- ( A e. InaccW -> A. z e. A E. w e. A z ~< w )
36	35	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> A. z e. A E. w e. A z ~< w )
37		breq1 4656	. . . . . . . . . . . . . . 15 |- ( z = ( aleph ` y ) -> ( z ~< w <-> ( aleph ` y ) ~< w ) )
38	37	rexbidv 3052	. . . . . . . . . . . . . 14 |- ( z = ( aleph ` y ) -> ( E. w e. A z ~< w <-> E. w e. A ( aleph ` y ) ~< w ) )
39	38	rspcva 3307	. . . . . . . . . . . . 13 |- ( ( ( aleph ` y ) e. A /\ A. z e. A E. w e. A z ~< w ) -> E. w e. A ( aleph ` y ) ~< w )
40	33, 36, 39	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> E. w e. A ( aleph ` y ) ~< w )
41	40	expr 643	. . . . . . . . . . 11 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ y e. On ) -> ( x = suc y -> E. w e. A ( aleph ` y ) ~< w ) )
42		iscard 8801	. . . . . . . . . . . . . . . . . . 19 |- ( ( card ` A ) = A <-> ( A e. On /\ A. w e. A w ~< A ) )
43	42	simprbi 480	. . . . . . . . . . . . . . . . . 18 |- ( ( card ` A ) = A -> A. w e. A w ~< A )
44		rsp 2929	. . . . . . . . . . . . . . . . . 18 |- ( A. w e. A w ~< A -> ( w e. A -> w ~< A ) )
45	1, 43, 44	3syl 18	. . . . . . . . . . . . . . . . 17 |- ( A e. InaccW -> ( w e. A -> w ~< A ) )
46	45	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( w e. A -> w ~< A ) )
47	32	breq2d 4665	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( w ~< A <-> w ~< ( aleph ` suc y ) ) )
48	46, 47	sylibd 229	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( w e. A -> w ~< ( aleph ` suc y ) ) )
49		alephnbtwn2 8895	. . . . . . . . . . . . . . . 16 |- -. ( ( aleph ` y ) ~< w /\ w ~< ( aleph ` suc y ) )
50		pm3.21 464	. . . . . . . . . . . . . . . 16 |- ( w ~< ( aleph ` suc y ) -> ( ( aleph ` y ) ~< w -> ( ( aleph ` y ) ~< w /\ w ~< ( aleph ` suc y ) ) ) )
51	49, 50	mtoi 190	. . . . . . . . . . . . . . 15 |- ( w ~< ( aleph ` suc y ) -> -. ( aleph ` y ) ~< w )
52	48, 51	syl6 35	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> ( w e. A -> -. ( aleph ` y ) ~< w ) )
53	52	imp 445	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) /\ w e. A ) -> -. ( aleph ` y ) ~< w )
54	53	nrexdv 3001	. . . . . . . . . . . 12 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ ( y e. On /\ x = suc y ) ) -> -. E. w e. A ( aleph ` y ) ~< w )
55	54	expr 643	. . . . . . . . . . 11 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ y e. On ) -> ( x = suc y -> -. E. w e. A ( aleph ` y ) ~< w ) )
56	41, 55	pm2.65d 187	. . . . . . . . . 10 |- ( ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) /\ y e. On ) -> -. x = suc y )
57	56	nrexdv 3001	. . . . . . . . 9 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> -. E. y e. On x = suc y )
58	57	pm2.21d 118	. . . . . . . 8 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( E. y e. On x = suc y -> Lim x ) )
59		simpr 477	. . . . . . . . 9 |- ( ( x e. _V /\ Lim x ) -> Lim x )
60	59	a1i 11	. . . . . . . 8 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( ( x e. _V /\ Lim x ) -> Lim x ) )
61	22, 58, 60	3jaod 1392	. . . . . . 7 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( ( x = (/) \/ E. y e. On x = suc y \/ ( x e. _V /\ Lim x ) ) -> Lim x ) )
62	12, 61	mpd 15	. . . . . 6 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> Lim x )
63	9, 62	jca 554	. . . . 5 |- ( ( ( A e. InaccW /\ A =/= om ) /\ ( x e. On /\ A = ( aleph ` x ) ) ) -> ( ( aleph ` x ) = A /\ Lim x ) )
64	63	ex 450	. . . 4 |- ( ( A e. InaccW /\ A =/= om ) -> ( ( x e. On /\ A = ( aleph ` x ) ) -> ( ( aleph ` x ) = A /\ Lim x ) ) )
65	64	eximdv 1846	. . 3 |- ( ( A e. InaccW /\ A =/= om ) -> ( E. x ( x e. On /\ A = ( aleph ` x ) ) -> E. x ( ( aleph ` x ) = A /\ Lim x ) ) )
66	7, 65	syl5bi 232	. 2 |- ( ( A e. InaccW /\ A =/= om ) -> ( E. x e. On A = ( aleph ` x ) -> E. x ( ( aleph ` x ) = A /\ Lim x ) ) )
67	6, 66	mpd 15	1 |- ( ( A e. InaccW /\ A =/= om ) -> E. x ( ( aleph ` x ) = A /\ Lim x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ` cfv 5888   omcom 7065   ~< csdm 7954   cardccrd 8761   alephcale 8762   cfccf 8763   InaccWcwina 9504

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  ax-inf2 8538

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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cf 8767  df-wina 9506

This theorem is referenced by:  winafp  9519