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Theorem hargch 9116
Description: If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 9122. (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 8094 . . . . . . . . . . . . . 14  |-  (har `  A )  e.  On
2 sdomdom 7615 . . . . . . . . . . . . . 14  |-  ( x 
~<  (har `  A )  ->  x  ~<_  (har `  A
) )
3 ondomen 8486 . . . . . . . . . . . . . 14  |-  ( ( (har `  A )  e.  On  /\  x  ~<_  (har
`  A ) )  ->  x  e.  dom  card )
41, 2, 3sylancr 676 . . . . . . . . . . . . 13  |-  ( x 
~<  (har `  A )  ->  x  e.  dom  card )
5 onenon 8401 . . . . . . . . . . . . . 14  |-  ( (har
`  A )  e.  On  ->  (har `  A
)  e.  dom  card )
61, 5ax-mp 5 . . . . . . . . . . . . 13  |-  (har `  A )  e.  dom  card
7 cardsdom2 8440 . . . . . . . . . . . . 13  |-  ( ( x  e.  dom  card  /\  (har `  A )  e.  dom  card )  ->  (
( card `  x )  e.  ( card `  (har `  A ) )  <->  x  ~<  (har
`  A ) ) )
84, 6, 7sylancl 675 . . . . . . . . . . . 12  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  e.  ( card `  (har `  A )
)  <->  x  ~<  (har `  A ) ) )
98ibir 250 . . . . . . . . . . 11  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  ( card `  (har `  A )
) )
10 harcard 8430 . . . . . . . . . . 11  |-  ( card `  (har `  A )
)  =  (har `  A )
119, 10syl6eleq 2559 . . . . . . . . . 10  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  (har `  A ) )
12 elharval 8096 . . . . . . . . . . 11  |-  ( (
card `  x )  e.  (har `  A )  <->  ( ( card `  x
)  e.  On  /\  ( card `  x )  ~<_  A ) )
1312simprbi 471 . . . . . . . . . 10  |-  ( (
card `  x )  e.  (har `  A )  ->  ( card `  x
)  ~<_  A )
1411, 13syl 17 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  ~<_  A )
15 cardid2 8405 . . . . . . . . . 10  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
16 domen1 7732 . . . . . . . . . 10  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  ~<_  A 
<->  x  ~<_  A ) )
174, 15, 163syl 18 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  ~<_  A  <->  x  ~<_  A ) )
1814, 17mpbid 215 . . . . . . . 8  |-  ( x 
~<  (har `  A )  ->  x  ~<_  A )
19 domnsym 7716 . . . . . . . 8  |-  ( x  ~<_  A  ->  -.  A  ~<  x )
2018, 19syl 17 . . . . . . 7  |-  ( x 
~<  (har `  A )  ->  -.  A  ~<  x
)
2120con2i 124 . . . . . 6  |-  ( A 
~<  x  ->  -.  x  ~<  (har `  A )
)
22 sdomen2 7735 . . . . . . 7  |-  ( (har
`  A )  ~~  ~P A  ->  ( x 
~<  (har `  A )  <->  x 
~<  ~P A ) )
2322notbid 301 . . . . . 6  |-  ( (har
`  A )  ~~  ~P A  ->  ( -.  x  ~<  (har `  A
)  <->  -.  x  ~<  ~P A ) )
2421, 23syl5ib 227 . . . . 5  |-  ( (har
`  A )  ~~  ~P A  ->  ( A 
~<  x  ->  -.  x  ~<  ~P A ) )
25 imnan 429 . . . . 5  |-  ( ( A  ~<  x  ->  -.  x  ~<  ~P A
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2624, 25sylib 201 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2726alrimiv 1781 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
2827olcd 400 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e.  Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
29 relen 7592 . . . . 5  |-  Rel  ~~
3029brrelex2i 4881 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  ~P A  e.  _V )
31 pwexb 6621 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
3230, 31sylibr 217 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. 
_V )
33 elgch 9065 . . 3  |-  ( A  e.  _V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
3432, 33syl 17 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e. GCH 
<->  ( A  e.  Fin  \/ 
A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) ) )
3528, 34mpbird 240 1  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. GCH )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376   A.wal 1450    e. wcel 1904   _Vcvv 3031   ~Pcpw 3942   class class class wbr 4395   dom cdm 4839   Oncon0 5430   ` cfv 5589    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587  harchar 8089   cardccrd 8387  GCHcgch 9063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043  df-har 8091  df-card 8391  df-gch 9064
This theorem is referenced by: (None)
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