Theorem gch2 8956

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 3487	. . 3 |- ran aleph C_ _V
2		sseq2 3489	. . 3 |- (GCH = _V -> ( ran aleph C_ GCH <-> ran aleph C_ _V ) )
3	1, 2	mpbiri 233	. 2 |- (GCH = _V -> ran aleph C_ GCH )
4		cardidm 8243	. . . . . . . 8 |- ( card ` ( card ` x ) ) = ( card ` x )
5		iscard3 8377	. . . . . . . 8 |- ( ( card ` ( card ` x ) ) = ( card ` x ) <-> ( card ` x ) e. ( om u. ran aleph ) )
6	4, 5	mpbi 208	. . . . . . 7 |- ( card ` x ) e. ( om u. ran aleph )
7		elun 3608	. . . . . . 7 |- ( ( card ` x ) e. ( om u. ran aleph ) <-> ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph ) )
8	6, 7	mpbi 208	. . . . . 6 |- ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph )
9		fingch 8904	. . . . . . . . 9 |- Fin C_ GCH
10		nnfi 7617	. . . . . . . . 9 |- ( ( card ` x ) e. om -> ( card ` x ) e. Fin )
11	9, 10	sseldi 3465	. . . . . . . 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 3461	. . . . . . 7 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. ran aleph -> ( card ` x ) e. GCH ) )
14	12, 13	jaod 380	. . . . . 6 |- ( ran aleph C_ GCH -> ( ( ( card ` x ) e. om \/ ( card ` x ) e. ran aleph ) -> ( card ` x ) e. GCH ) )
15	8, 14	mpi 17	. . . . 5 |- ( ran aleph C_ GCH -> ( card ` x ) e. GCH )
16		vex 3081	. . . . . . 7 |- x e. _V
17		alephon 8353	. . . . . . . . . . 11 |- ( aleph ` suc x ) e. On
18		simpr 461	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> x e. On )
19		simpl 457	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ran aleph C_ GCH )
20		alephfnon 8349	. . . . . . . . . . . . . 14 |- aleph Fn On
21		fnfvelrn 5952	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
22	20, 18, 21	sylancr 663	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
23	19, 22	sseldd 3468	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. GCH )
24		suceloni 6537	. . . . . . . . . . . . . . 15 |- ( x e. On -> suc x e. On )
25	24	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ran aleph C_ GCH /\ x e. On ) -> suc x e. On )
26		fnfvelrn 5952	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ suc x e. On ) -> ( aleph ` suc x ) e. ran aleph )
27	20, 25, 26	sylancr 663	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. ran aleph )
28	19, 27	sseldd 3468	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. GCH )
29		gchaleph2 8953	. . . . . . . . . . . 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 1219	. . . . . . . . . . 11 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
31		isnumi 8230	. . . . . . . . . . 11 |- ( ( ( aleph ` suc x ) e. On /\ ( aleph ` suc x ) ~~ ~P ( aleph ` x ) ) -> ~P ( aleph ` x ) e. dom card )
32	17, 30, 31	sylancr 663	. . . . . . . . . 10 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ~P ( aleph ` x ) e. dom card )
33	32	ralrimiva 2830	. . . . . . . . 9 |- ( ran aleph C_ GCH -> A. x e. On ~P ( aleph ` x ) e. dom card )
34		dfac12 8432	. . . . . . . . 9 |- (CHOICE <-> A. x e. On ~P ( aleph ` x ) e. dom card )
35	33, 34	sylibr 212	. . . . . . . 8 |- ( ran aleph C_ GCH -> CHOICE )
36		dfac10 8420	. . . . . . . 8 |- (CHOICE <-> dom card = _V )
37	35, 36	sylib 196	. . . . . . 7 |- ( ran aleph C_ GCH -> dom card = _V )
38	16, 37	syl5eleqr 2549	. . . . . 6 |- ( ran aleph C_ GCH -> x e. dom card )
39		cardid2 8237	. . . . . 6 |- ( x e. dom card -> ( card ` x ) ~~ x )
40		engch 8909	. . . . . 6 |- ( ( card ` x ) ~~ x -> ( ( card ` x ) e. GCH <-> x e. GCH ) )
41	38, 39, 40	3syl 20	. . . . 5 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. GCH <-> x e. GCH ) )
42	15, 41	mpbid 210	. . . 4 |- ( ran aleph C_ GCH -> x e. GCH )
43	16	a1i 11	. . . 4 |- ( ran aleph C_ GCH -> x e. _V )
44	42, 43	2thd 240	. . 3 |- ( ran aleph C_ GCH -> ( x e. GCH <-> x e. _V ) )
45	44	eqrdv 2451	. 2 |- ( ran aleph C_ GCH -> GCH = _V )
46	3, 45	impbii 188	1 |- (GCH = _V <-> ran aleph C_ GCH )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2799   _Vcvv 3078   u. cun 3437   C_ wss 3439   ~Pcpw 3971   class class class wbr 4403   Oncon0 4830   suc csuc 4832   dom cdm 4951   ran crn 4952   Fn wfn 5524   ` cfv 5529   omcom 6589   ~~ cen 7420   Fincfn 7423   cardccrd 8219   alephcale 8220  CHOICEwac 8399  GCHcgch 8901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-inf2 7961

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-har 7887  df-wdom 7888  df-cnf 7982  df-r1 8085  df-rank 8086  df-card 8223  df-aleph 8224  df-ac 8400  df-cda 8451  df-fin4 8570  df-gch 8902

This theorem is referenced by:  gch3  8957