Theorem hargch 9495

Description: If A + ~~ ~P A, then A is a GCH-set. The much simpler converse to gchhar 9501. (Contributed by Mario Carneiro, 2-Jun-2015.)

Assertion

Ref	Expression
hargch	|- ( (har ` A ) ~~ ~P A -> A e. GCH )

Proof of Theorem hargch

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		harcl 8466	. . . . . . . . . . . . . 14 |- (har ` A ) e. On
2		sdomdom 7983	. . . . . . . . . . . . . 14 |- ( x ~< (har ` A ) -> x ~<_ (har ` A ) )
3		ondomen 8860	. . . . . . . . . . . . . 14 |- ( ( (har ` A ) e. On /\ x ~<_ (har ` A ) ) -> x e. dom card )
4	1, 2, 3	sylancr 695	. . . . . . . . . . . . 13 |- ( x ~< (har ` A ) -> x e. dom card )
5		onenon 8775	. . . . . . . . . . . . . 14 |- ( (har ` A ) e. On -> (har ` A ) e. dom card )
6	1, 5	ax-mp 5	. . . . . . . . . . . . 13 |- (har ` A ) e. dom card
7		cardsdom2 8814	. . . . . . . . . . . . 13 |- ( ( x e. dom card /\ (har ` A ) e. dom card ) -> ( ( card ` x ) e. ( card ` (har ` A ) ) <-> x ~< (har ` A ) ) )
8	4, 6, 7	sylancl 694	. . . . . . . . . . . 12 |- ( x ~< (har ` A ) -> ( ( card ` x ) e. ( card ` (har ` A ) ) <-> x ~< (har ` A ) ) )
9	8	ibir 257	. . . . . . . . . . 11 |- ( x ~< (har ` A ) -> ( card ` x ) e. ( card ` (har ` A ) ) )
10		harcard 8804	. . . . . . . . . . 11 |- ( card ` (har ` A ) ) = (har ` A )
11	9, 10	syl6eleq 2711	. . . . . . . . . 10 |- ( x ~< (har ` A ) -> ( card ` x ) e. (har ` A ) )
12		elharval 8468	. . . . . . . . . . 11 |- ( ( card ` x ) e. (har ` A ) <-> ( ( card ` x ) e. On /\ ( card ` x ) ~<_ A ) )
13	12	simprbi 480	. . . . . . . . . 10 |- ( ( card ` x ) e. (har ` A ) -> ( card ` x ) ~<_ A )
14	11, 13	syl 17	. . . . . . . . 9 |- ( x ~< (har ` A ) -> ( card ` x ) ~<_ A )
15		cardid2 8779	. . . . . . . . . 10 |- ( x e. dom card -> ( card ` x ) ~~ x )
16		domen1 8102	. . . . . . . . . 10 |- ( ( card ` x ) ~~ x -> ( ( card ` x ) ~<_ A <-> x ~<_ A ) )
17	4, 15, 16	3syl 18	. . . . . . . . 9 |- ( x ~< (har ` A ) -> ( ( card ` x ) ~<_ A <-> x ~<_ A ) )
18	14, 17	mpbid 222	. . . . . . . 8 |- ( x ~< (har ` A ) -> x ~<_ A )
19		domnsym 8086	. . . . . . . 8 |- ( x ~<_ A -> -. A ~< x )
20	18, 19	syl 17	. . . . . . 7 |- ( x ~< (har ` A ) -> -. A ~< x )
21	20	con2i 134	. . . . . 6 |- ( A ~< x -> -. x ~< (har ` A ) )
22		sdomen2 8105	. . . . . . 7 |- ( (har ` A ) ~~ ~P A -> ( x ~< (har ` A ) <-> x ~< ~P A ) )
23	22	notbid 308	. . . . . 6 |- ( (har ` A ) ~~ ~P A -> ( -. x ~< (har ` A ) <-> -. x ~< ~P A ) )
24	21, 23	syl5ib 234	. . . . 5 |- ( (har ` A ) ~~ ~P A -> ( A ~< x -> -. x ~< ~P A ) )
25		imnan 438	. . . . 5 |- ( ( A ~< x -> -. x ~< ~P A ) <-> -. ( A ~< x /\ x ~< ~P A ) )
26	24, 25	sylib 208	. . . 4 |- ( (har ` A ) ~~ ~P A -> -. ( A ~< x /\ x ~< ~P A ) )
27	26	alrimiv 1855	. . 3 |- ( (har ` A ) ~~ ~P A -> A. x -. ( A ~< x /\ x ~< ~P A ) )
28	27	olcd 408	. 2 |- ( (har ` A ) ~~ ~P A -> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) )
29		relen 7960	. . . . 5 |- Rel ~~
30	29	brrelex2i 5159	. . . 4 |- ( (har ` A ) ~~ ~P A -> ~P A e. _V )
31		pwexb 6975	. . . 4 |- ( A e. _V <-> ~P A e. _V )
32	30, 31	sylibr 224	. . 3 |- ( (har ` A ) ~~ ~P A -> A e. _V )
33		elgch 9444	. . 3 |- ( A e. _V -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
34	32, 33	syl 17	. 2 |- ( (har ` A ) ~~ ~P A -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
35	28, 34	mpbird 247	1 |- ( (har ` A ) ~~ ~P A -> A e. GCH )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955  harchar 8461   cardccrd 8761  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

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-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-card 8765  df-gch 9443

This theorem is referenced by: (None)