Theorem gch2 9042

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 3509	. . 3 |- ran aleph C_ _V
2		sseq2 3511	. . 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 8331	. . . . . . . 8 |- ( card ` ( card ` x ) ) = ( card ` x )
5		iscard3 8465	. . . . . . . 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 3631	. . . . . . 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 8990	. . . . . . . . 9 |- Fin C_ GCH
10		nnfi 7703	. . . . . . . . 9 |- ( ( card ` x ) e. om -> ( card ` x ) e. Fin )
11	9, 10	sseldi 3487	. . . . . . . 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 3483	. . . . . . 7 |- ( ran aleph C_ GCH -> ( ( card ` x ) e. ran aleph -> ( card ` x ) e. GCH ) )
14	12, 13	jaod 378	. . . . . 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 3109	. . . . . . 7 |- x e. _V
17		alephon 8441	. . . . . . . . . . 11 |- ( aleph ` suc x ) e. On
18		simpr 459	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> x e. On )
19		simpl 455	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ran aleph C_ GCH )
20		alephfnon 8437	. . . . . . . . . . . . . 14 |- aleph Fn On
21		fnfvelrn 6004	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
22	20, 18, 21	sylancr 661	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. ran aleph )
23	19, 22	sseldd 3490	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` x ) e. GCH )
24		suceloni 6621	. . . . . . . . . . . . . . 15 |- ( x e. On -> suc x e. On )
25	24	adantl 464	. . . . . . . . . . . . . 14 |- ( ( ran aleph C_ GCH /\ x e. On ) -> suc x e. On )
26		fnfvelrn 6004	. . . . . . . . . . . . . 14 |- ( ( aleph Fn On /\ suc x e. On ) -> ( aleph ` suc x ) e. ran aleph )
27	20, 25, 26	sylancr 661	. . . . . . . . . . . . 13 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. ran aleph )
28	19, 27	sseldd 3490	. . . . . . . . . . . 12 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) e. GCH )
29		gchaleph2 9039	. . . . . . . . . . . 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 1226	. . . . . . . . . . 11 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
31		isnumi 8318	. . . . . . . . . . 11 |- ( ( ( aleph ` suc x ) e. On /\ ( aleph ` suc x ) ~~ ~P ( aleph ` x ) ) -> ~P ( aleph ` x ) e. dom card )
32	17, 30, 31	sylancr 661	. . . . . . . . . 10 |- ( ( ran aleph C_ GCH /\ x e. On ) -> ~P ( aleph ` x ) e. dom card )
33	32	ralrimiva 2868	. . . . . . . . 9 |- ( ran aleph C_ GCH -> A. x e. On ~P ( aleph ` x ) e. dom card )
34		dfac12 8520	. . . . . . . . 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 8508	. . . . . . . 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 8325	. . . . . 6 |- ( x e. dom card -> ( card ` x ) ~~ x )
40		engch 8995	. . . . . 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 366   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   u. cun 3459   C_ wss 3461   ~Pcpw 3999   class class class wbr 4439   Oncon0 4867   suc csuc 4869   dom cdm 4988   ran crn 4989   Fn wfn 5565   ` cfv 5570   omcom 6673   ~~ cen 7506   Fincfn 7509   cardccrd 8307   alephcale 8308  CHOICEwac 8487  GCHcgch 8987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-har 7976  df-wdom 7977  df-cnf 8070  df-r1 8173  df-rank 8174  df-card 8311  df-aleph 8312  df-ac 8488  df-cda 8539  df-fin4 8658  df-gch 8988

This theorem is referenced by:  gch3  9043