Theorem gch2 9065

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 3529	. . 3 |- ran aleph C_ _V
2		sseq2 3531	. . 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 8352	. . . . . . . 8 |- ( card ` ( card ` x ) ) = ( card ` x )
5		iscard3 8486	. . . . . . . 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 3650	. . . . . . 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 9013	. . . . . . . . 9 |- Fin C_ GCH
10		nnfi 7722	. . . . . . . . 9 |- ( ( card ` x ) e. om -> ( card ` x ) e. Fin )
11	9, 10	sseldi 3507	. . . . . . . 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 3503	. . . . . . 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 3121	. . . . . . 7 |- x e. _V
17		alephon 8462	. . . . . . . . . . 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 8458	. . . . . . . . . . . . . 14 |- aleph Fn On
21		fnfvelrn 6029	. . . . . . . . . . . . . 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 3510	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. GCH )
24		suceloni 6643	. . . . . . . . . . . . . . 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 6029	. . . . . . . . . . . . . 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 3510	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. GCH )
29		gchaleph2 9062	. . . . . . . . . . . 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 1228	. . . . . . . . . . 11 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
31		isnumi 8339	. . . . . . . . . . 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 2881	. . . . . . . . 9 |- ( ran aleph C_ GCH -> A. x e. On ~P ( aleph ` x ) e. dom card )
34		dfac12 8541	. . . . . . . . 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 8529	. . . . . . . 8 |- (CHOICE <-> dom card = _V )
37	35, 36	sylib 196	. . . . . . 7 |- ( ran aleph C_ GCH -> dom card = _V )
38	16, 37	syl5eleqr 2562	. . . . . 6 |- ( ran aleph C_ GCH -> x e. dom card )
39		cardid2 8346	. . . . . 6 |- ( x e. dom card -> ( card ` x ) ~~ x )
40		engch 9018	. . . . . 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 2464	. 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 1379   e. wcel 1767   A.wral 2817   _Vcvv 3118   u. cun 3479   C_ wss 3481   ~Pcpw 4016   class class class wbr 4453   Oncon0 4884   suc csuc 4886   dom cdm 5005   ran crn 5006   Fn wfn 5589   ` cfv 5594   omcom 6695   ~~ cen 7525   Fincfn 7528   cardccrd 8328   alephcale 8329  CHOICEwac 8508  GCHcgch 9010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-seqom 7125  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-oexp 7148  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-har 7996  df-wdom 7997  df-cnf 8091  df-r1 8194  df-rank 8195  df-card 8332  df-aleph 8333  df-ac 8509  df-cda 8560  df-fin4 8679  df-gch 9011

This theorem is referenced by:  gch3  9066