Theorem gchaleph 9493

Description: If ( aleph ` A ) is a GCH-set and its powerset is well-orderable, then the successor aleph ( aleph ` suc A ) is equinumerous to the powerset of ( aleph ` A ). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gchaleph	|- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( aleph ` suc A ) ~~ ~P ( aleph ` A ) )

Proof of Theorem gchaleph

Step	Hyp	Ref	Expression
1		alephsucpw2 8934	. . 3 |- -. ~P ( aleph ` A ) ~< ( aleph ` suc A )
2		alephon 8892	. . . . 5 |- ( aleph ` suc A ) e. On
3		onenon 8775	. . . . 5 |- ( ( aleph ` suc A ) e. On -> ( aleph ` suc A ) e. dom card )
4	2, 3	ax-mp 5	. . . 4 |- ( aleph ` suc A ) e. dom card
5		simp3 1063	. . . 4 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ~P ( aleph ` A ) e. dom card )
6		domtri2 8815	. . . 4 |- ( ( ( aleph ` suc A ) e. dom card /\ ~P ( aleph ` A ) e. dom card ) -> ( ( aleph ` suc A ) ~<_ ~P ( aleph ` A ) <-> -. ~P ( aleph ` A ) ~< ( aleph ` suc A ) ) )
7	4, 5, 6	sylancr 695	. . 3 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( ( aleph ` suc A ) ~<_ ~P ( aleph ` A ) <-> -. ~P ( aleph ` A ) ~< ( aleph ` suc A ) ) )
8	1, 7	mpbiri 248	. 2 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( aleph ` suc A ) ~<_ ~P ( aleph ` A ) )
9		fvex 6201	. . . . . . 7 |- ( aleph ` A ) e. _V
10		simp1 1061	. . . . . . . 8 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> A e. On )
11		alephgeom 8905	. . . . . . . 8 |- ( A e. On <-> om C_ ( aleph ` A ) )
12	10, 11	sylib 208	. . . . . . 7 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> om C_ ( aleph ` A ) )
13		ssdomg 8001	. . . . . . 7 |- ( ( aleph ` A ) e. _V -> ( om C_ ( aleph ` A ) -> om ~<_ ( aleph ` A ) ) )
14	9, 12, 13	mpsyl 68	. . . . . 6 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> om ~<_ ( aleph ` A ) )
15		domnsym 8086	. . . . . 6 |- ( om ~<_ ( aleph ` A ) -> -. ( aleph ` A ) ~< om )
16	14, 15	syl 17	. . . . 5 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> -. ( aleph ` A ) ~< om )
17		isfinite 8549	. . . . 5 |- ( ( aleph ` A ) e. Fin <-> ( aleph ` A ) ~< om )
18	16, 17	sylnibr 319	. . . 4 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> -. ( aleph ` A ) e. Fin )
19		simp2 1062	. . . . 5 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( aleph ` A ) e. GCH )
20		alephordilem1 8896	. . . . . 6 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
21	20	3ad2ant1 1082	. . . . 5 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( aleph ` A ) ~< ( aleph ` suc A ) )
22		gchi 9446	. . . . . 6 |- ( ( ( aleph ` A ) e. GCH /\ ( aleph ` A ) ~< ( aleph ` suc A ) /\ ( aleph ` suc A ) ~< ~P ( aleph ` A ) ) -> ( aleph ` A ) e. Fin )
23	22	3expia 1267	. . . . 5 |- ( ( ( aleph ` A ) e. GCH /\ ( aleph ` A ) ~< ( aleph ` suc A ) ) -> ( ( aleph ` suc A ) ~< ~P ( aleph ` A ) -> ( aleph ` A ) e. Fin ) )
24	19, 21, 23	syl2anc 693	. . . 4 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( ( aleph ` suc A ) ~< ~P ( aleph ` A ) -> ( aleph ` A ) e. Fin ) )
25	18, 24	mtod 189	. . 3 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> -. ( aleph ` suc A ) ~< ~P ( aleph ` A ) )
26		domtri2 8815	. . . 4 |- ( ( ~P ( aleph ` A ) e. dom card /\ ( aleph ` suc A ) e. dom card ) -> ( ~P ( aleph ` A ) ~<_ ( aleph ` suc A ) <-> -. ( aleph ` suc A ) ~< ~P ( aleph ` A ) ) )
27	5, 4, 26	sylancl 694	. . 3 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( ~P ( aleph ` A ) ~<_ ( aleph ` suc A ) <-> -. ( aleph ` suc A ) ~< ~P ( aleph ` A ) ) )
28	25, 27	mpbird 247	. 2 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ~P ( aleph ` A ) ~<_ ( aleph ` suc A ) )
29		sbth 8080	. 2 |- ( ( ( aleph ` suc A ) ~<_ ~P ( aleph ` A ) /\ ~P ( aleph ` A ) ~<_ ( aleph ` suc A ) ) -> ( aleph ` suc A ) ~~ ~P ( aleph ` A ) )
30	8, 28, 29	syl2anc 693	1 |- ( ( A e. On /\ ( aleph ` A ) e. GCH /\ ~P ( aleph ` A ) e. dom card ) -> ( aleph ` suc A ) ~~ ~P ( aleph ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   Oncon0 5723   suc csuc 5725   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761   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-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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-gch 9443

This theorem is referenced by:  gchaleph2  9494