Theorem gch2 9100

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 3452	. . 3 |- ran aleph C_ _V
2		sseq2 3454	. . 3 |- (GCH = _V -> ( ran aleph C_ GCH <-> ran aleph C_ _V ) )
3	1, 2	mpbiri 237	. 2 |- (GCH = _V -> ran aleph C_ GCH )
4		cardidm 8393	. . . . . . . 8 |- ( card ` ( card ` x ) ) = ( card ` x )
5		iscard3 8524	. . . . . . . 8 |- ( ( card ` ( card ` x ) ) = ( card ` x ) <-> ( card ` x ) e. ( om u. ran aleph ) )
6	4, 5	mpbi 212	. . . . . . 7 |- ( card ` x ) e. ( om u. ran aleph )
7		elun 3574	. . . . . . 7 |- ( ( card ` x ) e. ( om u. ran aleph ) <-> ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph ) )
8	6, 7	mpbi 212	. . . . . 6 |- ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph )
9		fingch 9048	. . . . . . . . 9 |- Fin C_ GCH
10		nnfi 7765	. . . . . . . . 9 |- ( ( card ` x ) e. om -> ( card ` x ) e. Fin )
11	9, 10	sseldi 3430	. . . . . . . 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 3426	. . . . . . 7 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. ran aleph -> ( card ` x ) e. GCH ) )
14	12, 13	jaod 382	. . . . . 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 3048	. . . . . . 7 |- x e. _V
17		alephon 8500	. . . . . . . . . . 11 |- ( aleph ` suc x ) e. On
18		simpr 463	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> x e. On )
19		simpl 459	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ran aleph C_ GCH )
20		alephfnon 8496	. . . . . . . . . . . . . 14 |- aleph Fn On
21		fnfvelrn 6019	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
22	20, 18, 21	sylancr 669	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
23	19, 22	sseldd 3433	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. GCH )
24		suceloni 6640	. . . . . . . . . . . . . . 15 |- ( x e. On -> suc x e. On )
25	24	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ran aleph C_ GCH /\ x e. On ) -> suc x e. On )
26		fnfvelrn 6019	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ suc x e. On ) -> ( aleph ` suc x ) e. ran aleph )
27	20, 25, 26	sylancr 669	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. ran aleph )
28	19, 27	sseldd 3433	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. GCH )
29		gchaleph2 9097	. . . . . . . . . . . 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 1268	. . . . . . . . . . 11 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
31		isnumi 8380	. . . . . . . . . . 11 |- ( ( ( aleph ` suc x ) e. On /\ ( aleph ` suc x ) ~~ ~P ( aleph ` x ) ) -> ~P ( aleph ` x ) e. dom card )
32	17, 30, 31	sylancr 669	. . . . . . . . . 10 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ~P ( aleph ` x ) e. dom card )
33	32	ralrimiva 2802	. . . . . . . . 9 |- ( ran aleph C_ GCH -> A. x e. On ~P ( aleph ` x ) e. dom card )
34		dfac12 8579	. . . . . . . . 9 |- (CHOICE <-> A. x e. On ~P ( aleph ` x ) e. dom card )
35	33, 34	sylibr 216	. . . . . . . 8 |- ( ran aleph C_ GCH -> CHOICE )
36		dfac10 8567	. . . . . . . 8 |- (CHOICE <-> dom card = _V )
37	35, 36	sylib 200	. . . . . . 7 |- ( ran aleph C_ GCH -> dom card = _V )
38	16, 37	syl5eleqr 2536	. . . . . 6 |- ( ran aleph C_ GCH -> x e. dom card )
39		cardid2 8387	. . . . . 6 |- ( x e. dom card -> ( card ` x ) ~~ x )
40		engch 9053	. . . . . 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 214	. . . 4 |- ( ran aleph C_ GCH -> x e. GCH )
43	16	a1i 11	. . . 4 |- ( ran aleph C_ GCH -> x e. _V )
44	42, 43	2thd 244	. . 3 |- ( ran aleph C_ GCH -> ( x e. GCH <-> x e. _V ) )
45	44	eqrdv 2449	. 2 |- ( ran aleph C_ GCH -> GCH = _V )
46	3, 45	impbii 191	1 |- (GCH = _V <-> ran aleph C_ GCH )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   _Vcvv 3045   u. cun 3402   C_ wss 3404   ~Pcpw 3951   class class class wbr 4402   dom cdm 4834   ran crn 4835   Oncon0 5423   suc csuc 5425   Fn wfn 5577   ` cfv 5582   omcom 6692   ~~ cen 7566   Fincfn 7569   cardccrd 8369   alephcale 8370  CHOICEwac 8546  GCHcgch 9045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-seqom 7165  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-oexp 7188  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-har 8073  df-wdom 8074  df-cnf 8167  df-r1 8235  df-rank 8236  df-card 8373  df-aleph 8374  df-ac 8547  df-cda 8598  df-fin4 8717  df-gch 9046

This theorem is referenced by:  gch3  9101