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Theorem hargch 8299
Description: If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 8293. (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 7275 . . . . . . . . . . . . . 14  |-  (har `  A )  e.  On
2 sdomdom 6889 . . . . . . . . . . . . . 14  |-  ( x 
~<  (har `  A )  ->  x  ~<_  (har `  A
) )
3 ondomen 7664 . . . . . . . . . . . . . 14  |-  ( ( (har `  A )  e.  On  /\  x  ~<_  (har
`  A ) )  ->  x  e.  dom  card )
41, 2, 3sylancr 644 . . . . . . . . . . . . 13  |-  ( x 
~<  (har `  A )  ->  x  e.  dom  card )
5 onenon 7582 . . . . . . . . . . . . . 14  |-  ( (har
`  A )  e.  On  ->  (har `  A
)  e.  dom  card )
61, 5ax-mp 8 . . . . . . . . . . . . 13  |-  (har `  A )  e.  dom  card
7 cardsdom2 7621 . . . . . . . . . . . . 13  |-  ( ( x  e.  dom  card  /\  (har `  A )  e.  dom  card )  ->  (
( card `  x )  e.  ( card `  (har `  A ) )  <->  x  ~<  (har
`  A ) ) )
84, 6, 7sylancl 643 . . . . . . . . . . . 12  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  e.  ( card `  (har `  A )
)  <->  x  ~<  (har `  A ) ) )
98ibir 233 . . . . . . . . . . 11  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  ( card `  (har `  A )
) )
10 harcard 7611 . . . . . . . . . . 11  |-  ( card `  (har `  A )
)  =  (har `  A )
119, 10syl6eleq 2373 . . . . . . . . . 10  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  (har `  A ) )
12 elharval 7277 . . . . . . . . . . 11  |-  ( (
card `  x )  e.  (har `  A )  <->  ( ( card `  x
)  e.  On  /\  ( card `  x )  ~<_  A ) )
1312simprbi 450 . . . . . . . . . 10  |-  ( (
card `  x )  e.  (har `  A )  ->  ( card `  x
)  ~<_  A )
1411, 13syl 15 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  ~<_  A )
15 cardid2 7586 . . . . . . . . . 10  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
16 domen1 7003 . . . . . . . . . 10  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  ~<_  A 
<->  x  ~<_  A ) )
174, 15, 163syl 18 . . . . . . . . 9  |-  ( x 
~<  (har `  A )  ->  ( ( card `  x
)  ~<_  A  <->  x  ~<_  A ) )
1814, 17mpbid 201 . . . . . . . 8  |-  ( x 
~<  (har `  A )  ->  x  ~<_  A )
19 domnsym 6987 . . . . . . . 8  |-  ( x  ~<_  A  ->  -.  A  ~<  x )
2018, 19syl 15 . . . . . . 7  |-  ( x 
~<  (har `  A )  ->  -.  A  ~<  x
)
2120con2i 112 . . . . . 6  |-  ( A 
~<  x  ->  -.  x  ~<  (har `  A )
)
22 sdomen2 7006 . . . . . . 7  |-  ( (har
`  A )  ~~  ~P A  ->  ( x 
~<  (har `  A )  <->  x 
~<  ~P A ) )
2322notbid 285 . . . . . 6  |-  ( (har
`  A )  ~~  ~P A  ->  ( -.  x  ~<  (har `  A
)  <->  -.  x  ~<  ~P A ) )
2421, 23syl5ib 210 . . . . 5  |-  ( (har
`  A )  ~~  ~P A  ->  ( A 
~<  x  ->  -.  x  ~<  ~P A ) )
25 imnan 411 . . . . 5  |-  ( ( A  ~<  x  ->  -.  x  ~<  ~P A
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2624, 25sylib 188 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )
2726alrimiv 1617 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
2827olcd 382 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e.  Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
29 relen 6868 . . . . 5  |-  Rel  ~~
3029brrelex2i 4730 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  ~P A  e.  _V )
31 pwexb 4564 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
3230, 31sylibr 203 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. 
_V )
33 elgch 8244 . . 3  |-  ( A  e.  _V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
3432, 33syl 15 . 2  |-  ( (har
`  A )  ~~  ~P A  ->  ( A  e. GCH 
<->  ( A  e.  Fin  \/ 
A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) ) )
3528, 34mpbird 223 1  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. GCH )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863  harchar 7270   cardccrd 7568  GCHcgch 8242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-card 7572  df-gch 8243
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