Theorem gch2 9497

Description: It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gch2	|- (GCH = _V <-> ran aleph C_ GCH )

Proof of Theorem gch2

Step	Hyp	Ref	Expression
1		ssv 3625	. . 3 |- ran aleph C_ _V
2		sseq2 3627	. . 3 |- (GCH = _V -> ( ran aleph C_ GCH <-> ran aleph C_ _V ) )
3	1, 2	mpbiri 248	. 2 |- (GCH = _V -> ran aleph C_ GCH )
4		cardidm 8785	. . . . . . . 8 |- ( card ` ( card ` x ) ) = ( card ` x )
5		iscard3 8916	. . . . . . . 8 |- ( ( card ` ( card ` x ) ) = ( card ` x ) <-> ( card ` x ) e. ( om u. ran aleph ) )
6	4, 5	mpbi 220	. . . . . . 7 |- ( card ` x ) e. ( om u. ran aleph )
7		elun 3753	. . . . . . 7 |- ( ( card ` x ) e. ( om u. ran aleph ) <-> ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph ) )
8	6, 7	mpbi 220	. . . . . 6 |- ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph )
9		fingch 9445	. . . . . . . . 9 |- Fin C_ GCH
10		nnfi 8153	. . . . . . . . 9 |- ( ( card ` x ) e. om -> ( card ` x ) e. Fin )
11	9, 10	sseldi 3601	. . . . . . . 8 |- ( ( card ` x ) e. om -> ( card ` x ) e. GCH )
12	11	a1i 11	. . . . . . 7 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. om -> ( card ` x ) e. GCH ) )
13		ssel 3597	. . . . . . 7 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. ran aleph -> ( card ` x ) e. GCH ) )
14	12, 13	jaod 395	. . . . . 6 |- ( ran aleph C_ GCH -> ( ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph ) -> ( card ` x ) e. GCH ) )
15	8, 14	mpi 20	. . . . 5 |- ( ran aleph C_ GCH -> ( card ` x ) e. GCH )
16		vex 3203	. . . . . . 7 |- x e. _V
17		alephon 8892	. . . . . . . . . . 11 |- ( aleph ` suc x ) e. On
18		simpr 477	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> x e. On )
19		simpl 473	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ran aleph C_ GCH )
20		alephfnon 8888	. . . . . . . . . . . . . 14 |- aleph Fn On
21		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
22	20, 18, 21	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
23	19, 22	sseldd 3604	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. GCH )
24		suceloni 7013	. . . . . . . . . . . . . . 15 |- ( x e. On -> suc x e. On )
25	24	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ran aleph C_ GCH /\ x e. On ) -> suc x e. On )
26		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ suc x e. On ) -> ( aleph ` suc x ) e. ran aleph )
27	20, 25, 26	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. ran aleph )
28	19, 27	sseldd 3604	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. GCH )
29		gchaleph2 9494	. . . . . . . . . . . 12 |- ( ( x e. On /\ ( aleph ` x ) e. GCH /\ ( aleph ` suc x ) e. GCH ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
30	18, 23, 28, 29	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
31		isnumi 8772	. . . . . . . . . . 11 |- ( ( ( aleph ` suc x ) e. On /\ ( aleph ` suc x ) ~~ ~P ( aleph ` x ) ) -> ~P ( aleph ` x ) e. dom card )
32	17, 30, 31	sylancr 695	. . . . . . . . . 10 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ~P ( aleph ` x ) e. dom card )
33	32	ralrimiva 2966	. . . . . . . . 9 |- ( ran aleph C_ GCH -> A. x e. On ~P ( aleph ` x ) e. dom card )
34		dfac12 8971	. . . . . . . . 9 |- (CHOICE <-> A. x e. On ~P ( aleph ` x ) e. dom card )
35	33, 34	sylibr 224	. . . . . . . 8 |- ( ran aleph C_ GCH -> CHOICE )
36		dfac10 8959	. . . . . . . 8 |- (CHOICE <-> dom card = _V )
37	35, 36	sylib 208	. . . . . . 7 |- ( ran aleph C_ GCH -> dom card = _V )
38	16, 37	syl5eleqr 2708	. . . . . 6 |- ( ran aleph C_ GCH -> x e. dom card )
39		cardid2 8779	. . . . . 6 |- ( x e. dom card -> ( card ` x ) ~~ x )
40		engch 9450	. . . . . 6 |- ( ( card ` x ) ~~ x -> ( ( card ` x ) e. GCH <-> x e. GCH ) )
41	38, 39, 40	3syl 18	. . . . 5 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. GCH <-> x e. GCH ) )
42	15, 41	mpbid 222	. . . 4 |- ( ran aleph C_ GCH -> x e. GCH )
43	16	a1i 11	. . . 4 |- ( ran aleph C_ GCH -> x e. _V )
44	42, 43	2thd 255	. . 3 |- ( ran aleph C_ GCH -> ( x e. GCH <-> x e. _V ) )
45	44	eqrdv 2620	. 2 |- ( ran aleph C_ GCH -> GCH = _V )
46	3, 45	impbii 199	1 |- (GCH = _V <-> ran aleph C_ GCH )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ran crn 5115   Oncon0 5723   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   ~~ cen 7952   Fincfn 7955   cardccrd 8761   alephcale 8762  CHOICEwac 8938  GCHcgch 9442

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-reg 8497  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-fal 1489  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-har 8463  df-wdom 8464  df-cnf 8559  df-r1 8627  df-rank 8628  df-card 8765  df-aleph 8766  df-ac 8939  df-cda 8990  df-fin4 9109  df-gch 9443

This theorem is referenced by:  gch3  9498