Theorem gch3 9498

Description: An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gch3	|- (GCH = _V <-> A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )

Proof of Theorem gch3

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( (GCH = _V /\ x e. On ) -> x e. On )
2		fvex 6201	. . . . 5 |- ( aleph ` x ) e. _V
3		simpl 473	. . . . 5 |- ( (GCH = _V /\ x e. On ) -> GCH = _V )
4	2, 3	syl5eleqr 2708	. . . 4 |- ( (GCH = _V /\ x e. On ) -> ( aleph ` x ) e. GCH )
5		fvex 6201	. . . . 5 |- ( aleph ` suc x ) e. _V
6	5, 3	syl5eleqr 2708	. . . 4 |- ( (GCH = _V /\ x e. On ) -> ( aleph ` suc x ) e. GCH )
7		gchaleph2 9494	. . . 4 |- ( ( x e. On /\ ( aleph ` x ) e. GCH /\ ( aleph ` suc x ) e. GCH ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
8	1, 4, 6, 7	syl3anc 1326	. . 3 |- ( (GCH = _V /\ x e. On ) -> ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
9	8	ralrimiva 2966	. 2 |- (GCH = _V -> A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )
10		alephgch 9496	. . . . . 6 |- ( ( aleph ` suc x ) ~~ ~P ( aleph ` x ) -> ( aleph ` x ) e. GCH )
11	10	ralimi 2952	. . . . 5 |- ( A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) -> A. x e. On ( aleph ` x ) e. GCH )
12		alephfnon 8888	. . . . . 6 |- aleph Fn On
13		ffnfv 6388	. . . . . 6 |- ( aleph : On -->GCH <-> ( aleph Fn On /\ A. x e. On ( aleph ` x ) e. GCH ) )
14	12, 13	mpbiran 953	. . . . 5 |- ( aleph : On -->GCH <-> A. x e. On ( aleph ` x ) e. GCH )
15	11, 14	sylibr 224	. . . 4 |- ( A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) -> aleph : On -->GCH )
16		df-f 5892	. . . . 5 |- ( aleph : On -->GCH <-> ( aleph Fn On /\ ran aleph C_ GCH ) )
17	12, 16	mpbiran 953	. . . 4 |- ( aleph : On -->GCH <-> ran aleph C_ GCH )
18	15, 17	sylib 208	. . 3 |- ( A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) -> ran aleph C_ GCH )
19		gch2 9497	. . 3 |- (GCH = _V <-> ran aleph C_ GCH )
20	18, 19	sylibr 224	. 2 |- ( A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) -> GCH = _V )
21	9, 20	impbii 199	1 |- (GCH = _V <-> A. x e. On ( aleph ` suc x ) ~~ ~P ( aleph ` x ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ran crn 5115   Oncon0 5723   suc csuc 5725   Fn wfn 5883   -->wf 5884   ` cfv 5888   ~~ cen 7952   alephcale 8762  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: (None)