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Theorem hargch 9103
Description: If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 9109. (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 8081 . . . . . . . . . . . . . 14  |-  (har `  A )  e.  On
2 sdomdom 7602 . . . . . . . . . . . . . 14  |-  ( x 
~<  (har `  A )  ->  x  ~<_  (har `  A
) )
3 ondomen 8473 . . . . . . . . . . . . . 14  |-  ( ( (har `  A )  e.  On  /\  x  ~<_  (har
`  A ) )  ->  x  e.  dom  card )
41, 2, 3sylancr 670 . . . . . . . . . . . . 13  |-  ( x 
~<  (har `  A )  ->  x  e.  dom  card )
5 onenon 8388 . . . . . . . . . . . . . 14  |-  ( (har
`  A )  e.  On  ->  (har `  A
)  e.  dom  card )
61, 5ax-mp 5 . . . . . . . . . . . . 13  |-  (har `  A )  e.  dom  card
7 cardsdom2 8427 . . . . . . . . . . . . 13  |-  ( ( x  e.  dom  card  /\  (har `  A )  e.  dom  card )  ->  (
( card `  x )  e.  ( card `  (har `  A ) )  <->  x  ~<  (har
`  A ) ) )
84, 6, 7sylancl 669 . . . . . . . . . . . 12  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  e.  ( card `  (har `  A )
)  <->  x  ~<  (har `  A ) ) )
98ibir 246 . . . . . . . . . . 11  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  ( card `  (har `  A )
) )
10 harcard 8417 . . . . . . . . . . 11  |-  ( card `  (har `  A )
)  =  (har `  A )
119, 10syl6eleq 2541 . . . . . . . . . 10  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  (har `  A ) )
12 elharval 8083 . . . . . . . . . . 11  |-  ( (
card `  x )  e.  (har `  A )  <->  ( ( card `  x
)  e.  On  /\  ( card `  x )  ~<_  A ) )
1312simprbi 466 . . . . . . . . . 10  |-  ( (
card `  x )  e.  (har `  A )  ->  ( card `  x
)  ~<_  A )
1411, 13syl 17 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  ~<_  A )
15 cardid2 8392 . . . . . . . . . 10  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
16 domen1 7719 . . . . . . . . . 10  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  ~<_  A 
<->  x  ~<_  A ) )
174, 15, 163syl 18 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  ~<_  A  <->  x  ~<_  A ) )
1814, 17mpbid 214 . . . . . . . 8  |-  ( x 
~<  (har `  A )  ->  x  ~<_  A )
19 domnsym 7703 . . . . . . . 8  |-  ( x  ~<_  A  ->  -.  A  ~<  x )
2018, 19syl 17 . . . . . . 7  |-  ( x 
~<  (har `  A )  ->  -.  A  ~<  x
)
2120con2i 124 . . . . . 6  |-  ( A 
~<  x  ->  -.  x  ~<  (har `  A )
)
22 sdomen2 7722 . . . . . . 7  |-  ( (har
`  A )  ~~  ~P A  ->  ( x 
~<  (har `  A )  <->  x 
~<  ~P A ) )
2322notbid 296 . . . . . 6  |-  ( (har
`  A )  ~~  ~P A  ->  ( -.  x  ~<  (har `  A
)  <->  -.  x  ~<  ~P A ) )
2421, 23syl5ib 223 . . . . 5  |-  ( (har
`  A )  ~~  ~P A  ->  ( A 
~<  x  ->  -.  x  ~<  ~P A ) )
25 imnan 424 . . . . 5  |-  ( ( A  ~<  x  ->  -.  x  ~<  ~P A
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2624, 25sylib 200 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2726alrimiv 1775 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
2827olcd 395 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e.  Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
29 relen 7579 . . . . 5  |-  Rel  ~~
3029brrelex2i 4879 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  ~P A  e.  _V )
31 pwexb 6607 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
3230, 31sylibr 216 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. 
_V )
33 elgch 9052 . . 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 236 1  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. GCH )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371   A.wal 1444    e. wcel 1889   _Vcvv 3047   ~Pcpw 3953   class class class wbr 4405   dom cdm 4837   Oncon0 5426   ` cfv 5585    ~~ cen 7571    ~<_ cdom 7572    ~< csdm 7573   Fincfn 7574  harchar 8076   cardccrd 8374  GCHcgch 9050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-wrecs 7033  df-recs 7095  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-oi 8030  df-har 8078  df-card 8378  df-gch 9051
This theorem is referenced by: (None)
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