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Theorem gchpwdom 8296
Description: A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchpwdom  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  <->  ~P A  ~<_  B ) )

Proof of Theorem gchpwdom
StepHypRef Expression
1 simpl2 959 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  e. GCH )
2 pwexg 4194 . . . . . . 7  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
31, 2syl 15 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  e.  _V )
4 simpl3 960 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  e. GCH )
5 cdadom3 7814 . . . . . 6  |-  ( ( ~P A  e.  _V  /\  B  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  B ) )
63, 4, 5syl2anc 642 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  ( ~P A  +c  B
) )
7 domen2 7004 . . . . 5  |-  ( B 
~~  ( ~P A  +c  B )  ->  ( ~P A  ~<_  B  <->  ~P A  ~<_  ( ~P A  +c  B
) ) )
86, 7syl5ibrcom 213 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~~  ( ~P A  +c  B )  ->  ~P A  ~<_  B ) )
9 cdacomen 7807 . . . . . . 7  |-  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B )
10 entr 6913 . . . . . . 7  |-  ( ( ( B  +c  ~P A )  ~~  ( ~P A  +c  B
)  /\  ( ~P A  +c  B )  ~~  ~P B )  ->  ( B  +c  ~P A ) 
~~  ~P B )
119, 10mpan 651 . . . . . 6  |-  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ( B  +c  ~P A )  ~~  ~P B )
12 ensym 6910 . . . . . 6  |-  ( ( B  +c  ~P A
)  ~~  ~P B  ->  ~P B  ~~  ( B  +c  ~P A ) )
13 endom 6888 . . . . . 6  |-  ( ~P B  ~~  ( B  +c  ~P A )  ->  ~P B  ~<_  ( B  +c  ~P A
) )
1411, 12, 133syl 18 . . . . 5  |-  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ~P B  ~<_  ( B  +c  ~P A ) )
15 domsdomtr 6996 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  ~<  B )  ->  om  ~<  B )
16153ad2antl1 1117 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<  B )
17 sdomnsym 6986 . . . . . . . . . 10  |-  ( om 
~<  B  ->  -.  B  ~<  om )
1816, 17syl 15 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  ~<  om )
19 isfinite 7353 . . . . . . . . 9  |-  ( B  e.  Fin  <->  B  ~<  om )
2018, 19sylnibr 296 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  e.  Fin )
21 gchcdaidm 8290 . . . . . . . 8  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  B
)  ~~  B )
224, 20, 21syl2anc 642 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  B )  ~~  B
)
23 pwen 7034 . . . . . . 7  |-  ( ( B  +c  B ) 
~~  B  ->  ~P ( B  +c  B
)  ~~  ~P B
)
24 domen1 7003 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~~  ~P B  ->  ( ~P ( B  +c  B )  ~<_  ( B  +c  ~P A
)  <->  ~P B  ~<_  ( B  +c  ~P A ) ) )
2522, 23, 243syl 18 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  <->  ~P B  ~<_  ( B  +c  ~P A
) ) )
26 pwcdadom 7842 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P B  ~<_  ~P A )
27 canth2g 7015 . . . . . . . . 9  |-  ( B  e. GCH  ->  B  ~<  ~P B
)
28 sdomdomtr 6994 . . . . . . . . . 10  |-  ( ( B  ~<  ~P B  /\  ~P B  ~<_  ~P A
)  ->  B  ~<  ~P A )
2928ex 423 . . . . . . . . 9  |-  ( B 
~<  ~P B  ->  ( ~P B  ~<_  ~P A  ->  B  ~<  ~P A
) )
304, 27, 293syl 18 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ~P A  ->  B  ~<  ~P A ) )
31 gchi 8246 . . . . . . . . . 10  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
32313expia 1153 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
33323ad2antl2 1118 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin ) )
34 isfinite 7353 . . . . . . . . 9  |-  ( A  e.  Fin  <->  A  ~<  om )
35 simpl1 958 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<_  A )
36 domnsym 6987 . . . . . . . . . . 11  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
3735, 36syl 15 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  A  ~<  om )
3837pm2.21d 98 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  ~<  om  ->  ~P A  ~<_  B ) )
3934, 38syl5bi 208 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  e.  Fin  ->  ~P A  ~<_  B ) )
4030, 33, 393syld 51 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ~P A  ->  ~P A  ~<_  B ) )
4126, 40syl5 28 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P A  ~<_  B ) )
4225, 41sylbird 226 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ( B  +c  ~P A )  ->  ~P A  ~<_  B ) )
4314, 42syl5 28 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ~P A  ~<_  B ) )
44 cdadom3 7814 . . . . . . 7  |-  ( ( B  e. GCH  /\  ~P A  e.  _V )  ->  B  ~<_  ( B  +c  ~P A ) )
454, 3, 44syl2anc 642 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( B  +c  ~P A ) )
46 domentr 6920 . . . . . 6  |-  ( ( B  ~<_  ( B  +c  ~P A )  /\  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B ) )  ->  B  ~<_  ( ~P A  +c  B ) )
4745, 9, 46sylancl 643 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( ~P A  +c  B ) )
48 sdomdom 6889 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  ~<_  B )
4948adantl 452 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  ~<_  B )
50 pwdom 7013 . . . . . . . 8  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
51 cdadom1 7812 . . . . . . . 8  |-  ( ~P A  ~<_  ~P B  ->  ( ~P A  +c  B
)  ~<_  ( ~P B  +c  B ) )
5249, 50, 513syl 18 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ( ~P B  +c  B
) )
534, 27syl 15 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<  ~P B )
54 sdomdom 6889 . . . . . . . 8  |-  ( B 
~<  ~P B  ->  B  ~<_  ~P B )
55 cdadom2 7813 . . . . . . . 8  |-  ( B  ~<_  ~P B  ->  ( ~P B  +c  B
)  ~<_  ( ~P B  +c  ~P B ) )
5653, 54, 553syl 18 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  B )  ~<_  ( ~P B  +c  ~P B ) )
57 domtr 6914 . . . . . . 7  |-  ( ( ( ~P A  +c  B )  ~<_  ( ~P B  +c  B )  /\  ( ~P B  +c  B )  ~<_  ( ~P B  +c  ~P B
) )  ->  ( ~P A  +c  B
)  ~<_  ( ~P B  +c  ~P B ) )
5852, 56, 57syl2anc 642 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ( ~P B  +c  ~P B ) )
59 pwcda1 7820 . . . . . . . 8  |-  ( B  e. GCH  ->  ( ~P B  +c  ~P B )  ~~  ~P ( B  +c  1o ) )
604, 59syl 15 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P ( B  +c  1o ) )
61 gchcda1 8278 . . . . . . . . 9  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  1o )  ~~  B )
624, 20, 61syl2anc 642 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  1o )  ~~  B
)
63 pwen 7034 . . . . . . . 8  |-  ( ( B  +c  1o ) 
~~  B  ->  ~P ( B  +c  1o )  ~~  ~P B )
6462, 63syl 15 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P ( B  +c  1o )  ~~  ~P B )
65 entr 6913 . . . . . . 7  |-  ( ( ( ~P B  +c  ~P B )  ~~  ~P ( B  +c  1o )  /\  ~P ( B  +c  1o )  ~~  ~P B )  ->  ( ~P B  +c  ~P B
)  ~~  ~P B
)
6660, 64, 65syl2anc 642 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P B )
67 domentr 6920 . . . . . 6  |-  ( ( ( ~P A  +c  B )  ~<_  ( ~P B  +c  ~P B
)  /\  ( ~P B  +c  ~P B ) 
~~  ~P B )  -> 
( ~P A  +c  B )  ~<_  ~P B
)
6858, 66, 67syl2anc 642 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ~P B )
69 gchor 8249 . . . . 5  |-  ( ( ( B  e. GCH  /\  -.  B  e.  Fin )  /\  ( B  ~<_  ( ~P A  +c  B
)  /\  ( ~P A  +c  B )  ~<_  ~P B ) )  -> 
( B  ~~  ( ~P A  +c  B
)  \/  ( ~P A  +c  B ) 
~~  ~P B ) )
704, 20, 47, 68, 69syl22anc 1183 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~~  ( ~P A  +c  B )  \/  ( ~P A  +c  B
)  ~~  ~P B
) )
718, 43, 70mpjaod 370 . . 3  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  B )
7271ex 423 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  ->  ~P A  ~<_  B )
)
73 reldom 6869 . . . . 5  |-  Rel  ~<_
7473brrelexi 4729 . . . 4  |-  ( ~P A  ~<_  B  ->  ~P A  e.  _V )
75 pwexb 4564 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
76 canth2g 7015 . . . . 5  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7775, 76sylbir 204 . . . 4  |-  ( ~P A  e.  _V  ->  A 
~<  ~P A )
7874, 77syl 15 . . 3  |-  ( ~P A  ~<_  B  ->  A  ~<  ~P A )
79 sdomdomtr 6994 . . 3  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<  B )
8078, 79mpancom 650 . 2  |-  ( ~P A  ~<_  B  ->  A  ~<  B )
8172, 80impbid1 194 1  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  <->  ~P A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023   omcom 4656  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863    +c ccda 7793  GCHcgch 8242
This theorem is referenced by:  gchaleph2  8298  gchina  8321
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  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  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-om 4657  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-wdom 7273  df-cnf 7363  df-card 7572  df-cda 7794  df-fin4 7913  df-gch 8243
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