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Theorem hargch 9068
Description: If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 9074. (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.
StepHypRef Expression
1 harcl 8005 . . . . . . . . . . . . . 14  |-  (har `  A )  e.  On
2 sdomdom 7562 . . . . . . . . . . . . . 14  |-  ( x 
~<  (har `  A )  ->  x  ~<_  (har `  A
) )
3 ondomen 8435 . . . . . . . . . . . . . 14  |-  ( ( (har `  A )  e.  On  /\  x  ~<_  (har
`  A ) )  ->  x  e.  dom  card )
41, 2, 3sylancr 663 . . . . . . . . . . . . 13  |-  ( x 
~<  (har `  A )  ->  x  e.  dom  card )
5 onenon 8347 . . . . . . . . . . . . . 14  |-  ( (har
`  A )  e.  On  ->  (har `  A
)  e.  dom  card )
61, 5ax-mp 5 . . . . . . . . . . . . 13  |-  (har `  A )  e.  dom  card
7 cardsdom2 8386 . . . . . . . . . . . . 13  |-  ( ( x  e.  dom  card  /\  (har `  A )  e.  dom  card )  ->  (
( card `  x )  e.  ( card `  (har `  A ) )  <->  x  ~<  (har
`  A ) ) )
84, 6, 7sylancl 662 . . . . . . . . . . . 12  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  e.  ( card `  (har `  A )
)  <->  x  ~<  (har `  A ) ) )
98ibir 242 . . . . . . . . . . 11  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  ( card `  (har `  A )
) )
10 harcard 8376 . . . . . . . . . . 11  |-  ( card `  (har `  A )
)  =  (har `  A )
119, 10syl6eleq 2555 . . . . . . . . . 10  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  (har `  A ) )
12 elharval 8007 . . . . . . . . . . 11  |-  ( (
card `  x )  e.  (har `  A )  <->  ( ( card `  x
)  e.  On  /\  ( card `  x )  ~<_  A ) )
1312simprbi 464 . . . . . . . . . 10  |-  ( (
card `  x )  e.  (har `  A )  ->  ( card `  x
)  ~<_  A )
1411, 13syl 16 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  ~<_  A )
15 cardid2 8351 . . . . . . . . . 10  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
16 domen1 7678 . . . . . . . . . 10  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  ~<_  A 
<->  x  ~<_  A ) )
174, 15, 163syl 20 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  ~<_  A  <->  x  ~<_  A ) )
1814, 17mpbid 210 . . . . . . . 8  |-  ( x 
~<  (har `  A )  ->  x  ~<_  A )
19 domnsym 7662 . . . . . . . 8  |-  ( x  ~<_  A  ->  -.  A  ~<  x )
2018, 19syl 16 . . . . . . 7  |-  ( x 
~<  (har `  A )  ->  -.  A  ~<  x
)
2120con2i 120 . . . . . 6  |-  ( A 
~<  x  ->  -.  x  ~<  (har `  A )
)
22 sdomen2 7681 . . . . . . 7  |-  ( (har
`  A )  ~~  ~P A  ->  ( x 
~<  (har `  A )  <->  x 
~<  ~P A ) )
2322notbid 294 . . . . . 6  |-  ( (har
`  A )  ~~  ~P A  ->  ( -.  x  ~<  (har `  A
)  <->  -.  x  ~<  ~P A ) )
2421, 23syl5ib 219 . . . . 5  |-  ( (har
`  A )  ~~  ~P A  ->  ( A 
~<  x  ->  -.  x  ~<  ~P A ) )
25 imnan 422 . . . . 5  |-  ( ( A  ~<  x  ->  -.  x  ~<  ~P A
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2624, 25sylib 196 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2726alrimiv 1720 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
2827olcd 393 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e.  Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
29 relen 7540 . . . . 5  |-  Rel  ~~
3029brrelex2i 5050 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  ~P A  e.  _V )
31 pwexb 6610 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
3230, 31sylibr 212 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. 
_V )
33 elgch 9017 . . 3  |-  ( A  e.  _V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
3432, 33syl 16 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e. GCH 
<->  ( A  e.  Fin  \/ 
A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) ) )
3528, 34mpbird 232 1  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. GCH )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1393    e. wcel 1819   _Vcvv 3109   ~Pcpw 4015   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ` cfv 5594    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535  harchar 8000   cardccrd 8333  GCHcgch 9015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-oi 7953  df-har 8002  df-card 8337  df-gch 9016
This theorem is referenced by: (None)
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