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Theorem hargch 9040
Description: If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 9046. (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 7976 . . . . . . . . . . . . . 14  |-  (har `  A )  e.  On
2 sdomdom 7533 . . . . . . . . . . . . . 14  |-  ( x 
~<  (har `  A )  ->  x  ~<_  (har `  A
) )
3 ondomen 8407 . . . . . . . . . . . . . 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 8319 . . . . . . . . . . . . . 14  |-  ( (har
`  A )  e.  On  ->  (har `  A
)  e.  dom  card )
61, 5ax-mp 5 . . . . . . . . . . . . 13  |-  (har `  A )  e.  dom  card
7 cardsdom2 8358 . . . . . . . . . . . . 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 8348 . . . . . . . . . . 11  |-  ( card `  (har `  A )
)  =  (har `  A )
119, 10syl6eleq 2558 . . . . . . . . . 10  |-  ( x 
~<  (har `  A )  ->  ( card `  x
)  e.  (har `  A ) )
12 elharval 7978 . . . . . . . . . . 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 8323 . . . . . . . . . 10  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
16 domen1 7649 . . . . . . . . . 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 7633 . . . . . . . 8  |-  ( x  ~<_  A  ->  -.  A  ~<  x )
2018, 19syl 16 . . . . . . 7  |-  ( x 
~<  (har `  A )  ->  -.  A  ~<  x
)
2120con2i 120 . . . . . 6  |-  ( A 
~<  x  ->  -.  x  ~<  (har `  A )
)
22 sdomen2 7652 . . . . . . 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 1690 . . 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 7511 . . . . 5  |-  Rel  ~~
3029brrelex2i 5033 . . . 4  |-  ( (har
`  A )  ~~  ~P A  ->  ~P A  e.  _V )
31 pwexb 6582 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
3230, 31sylibr 212 . . 3  |-  ( (har
`  A )  ~~  ~P A  ->  A  e. 
_V )
33 elgch 8989 . . 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 1372    e. wcel 1762   _Vcvv 3106   ~Pcpw 4003   class class class wbr 4440   Oncon0 4871   dom cdm 4992   ` cfv 5579    ~~ cen 7503    ~<_ cdom 7504    ~< csdm 7505   Fincfn 7506  harchar 7971   cardccrd 8305  GCHcgch 8987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-recs 7032  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-oi 7924  df-har 7973  df-card 8309  df-gch 8988
This theorem is referenced by: (None)
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