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Theorem gchpwdom 8229
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 964 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  e. GCH )
2 pwexg 4132 . . . . . . 7  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
31, 2syl 17 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  e.  _V )
4 simpl3 965 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  e. GCH )
5 cdadom3 7747 . . . . . 6  |-  ( ( ~P A  e.  _V  /\  B  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  B ) )
63, 4, 5syl2anc 645 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  ( ~P A  +c  B
) )
7 domen2 6937 . . . . 5  |-  ( B 
~~  ( ~P A  +c  B )  ->  ( ~P A  ~<_  B  <->  ~P A  ~<_  ( ~P A  +c  B
) ) )
86, 7syl5ibrcom 215 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~~  ( ~P A  +c  B )  ->  ~P A  ~<_  B ) )
9 cdacomen 7740 . . . . . . 7  |-  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B )
10 entr 6846 . . . . . . 7  |-  ( ( ( B  +c  ~P A )  ~~  ( ~P A  +c  B
)  /\  ( ~P A  +c  B )  ~~  ~P B )  ->  ( B  +c  ~P A ) 
~~  ~P B )
119, 10mpan 654 . . . . . 6  |-  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ( B  +c  ~P A )  ~~  ~P B )
12 ensym 6843 . . . . . 6  |-  ( ( B  +c  ~P A
)  ~~  ~P B  ->  ~P B  ~~  ( B  +c  ~P A ) )
13 endom 6821 . . . . . 6  |-  ( ~P B  ~~  ( B  +c  ~P A )  ->  ~P B  ~<_  ( B  +c  ~P A
) )
1411, 12, 133syl 20 . . . . 5  |-  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ~P B  ~<_  ( B  +c  ~P A ) )
15 domsdomtr 6929 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  ~<  B )  ->  om  ~<  B )
16153ad2antl1 1122 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<  B )
17 sdomnsym 6919 . . . . . . . . . 10  |-  ( om 
~<  B  ->  -.  B  ~<  om )
1816, 17syl 17 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  ~<  om )
19 isfinite 7286 . . . . . . . . 9  |-  ( B  e.  Fin  <->  B  ~<  om )
2018, 19sylnibr 298 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  e.  Fin )
21 gchcdaidm 8223 . . . . . . . 8  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  B
)  ~~  B )
224, 20, 21syl2anc 645 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  B )  ~~  B
)
23 pwen 6967 . . . . . . 7  |-  ( ( B  +c  B ) 
~~  B  ->  ~P ( B  +c  B
)  ~~  ~P B
)
24 domen1 6936 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~~  ~P B  ->  ( ~P ( B  +c  B )  ~<_  ( B  +c  ~P A
)  <->  ~P B  ~<_  ( B  +c  ~P A ) ) )
2522, 23, 243syl 20 . . . . . 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 7775 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P B  ~<_  ~P A )
27 canth2g 6948 . . . . . . . . 9  |-  ( B  e. GCH  ->  B  ~<  ~P B
)
28 sdomdomtr 6927 . . . . . . . . . 10  |-  ( ( B  ~<  ~P B  /\  ~P B  ~<_  ~P A
)  ->  B  ~<  ~P A )
2928ex 425 . . . . . . . . 9  |-  ( B 
~<  ~P B  ->  ( ~P B  ~<_  ~P A  ->  B  ~<  ~P A
) )
304, 27, 293syl 20 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ~P A  ->  B  ~<  ~P A ) )
31 gchi 8179 . . . . . . . . . 10  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
32313expia 1158 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
33323ad2antl2 1123 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin ) )
34 isfinite 7286 . . . . . . . . 9  |-  ( A  e.  Fin  <->  A  ~<  om )
35 simpl1 963 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<_  A )
36 domnsym 6920 . . . . . . . . . . 11  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
3735, 36syl 17 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  A  ~<  om )
3837pm2.21d 100 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  ~<  om  ->  ~P A  ~<_  B ) )
3934, 38syl5bi 210 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  e.  Fin  ->  ~P A  ~<_  B ) )
4030, 33, 393syld 53 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ~P A  ->  ~P A  ~<_  B ) )
4126, 40syl5 30 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P A  ~<_  B ) )
4225, 41sylbird 228 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ( B  +c  ~P A )  ->  ~P A  ~<_  B ) )
4314, 42syl5 30 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ~P A  ~<_  B ) )
44 cdadom3 7747 . . . . . . 7  |-  ( ( B  e. GCH  /\  ~P A  e.  _V )  ->  B  ~<_  ( B  +c  ~P A ) )
454, 3, 44syl2anc 645 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( B  +c  ~P A ) )
46 domentr 6853 . . . . . 6  |-  ( ( B  ~<_  ( B  +c  ~P A )  /\  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B ) )  ->  B  ~<_  ( ~P A  +c  B ) )
4745, 9, 46sylancl 646 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( ~P A  +c  B ) )
48 sdomdom 6822 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  ~<_  B )
4948adantl 454 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  ~<_  B )
50 pwdom 6946 . . . . . . . 8  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
51 cdadom1 7745 . . . . . . . 8  |-  ( ~P A  ~<_  ~P B  ->  ( ~P A  +c  B
)  ~<_  ( ~P B  +c  B ) )
5249, 50, 513syl 20 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ( ~P B  +c  B
) )
534, 27syl 17 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<  ~P B )
54 sdomdom 6822 . . . . . . . 8  |-  ( B 
~<  ~P B  ->  B  ~<_  ~P B )
55 cdadom2 7746 . . . . . . . 8  |-  ( B  ~<_  ~P B  ->  ( ~P B  +c  B
)  ~<_  ( ~P B  +c  ~P B ) )
5653, 54, 553syl 20 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  B )  ~<_  ( ~P B  +c  ~P B ) )
57 domtr 6847 . . . . . . 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 645 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ( ~P B  +c  ~P B ) )
59 pwcda1 7753 . . . . . . . 8  |-  ( B  e. GCH  ->  ( ~P B  +c  ~P B )  ~~  ~P ( B  +c  1o ) )
604, 59syl 17 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P ( B  +c  1o ) )
61 gchcda1 8211 . . . . . . . . 9  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  1o )  ~~  B )
624, 20, 61syl2anc 645 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  1o )  ~~  B
)
63 pwen 6967 . . . . . . . 8  |-  ( ( B  +c  1o ) 
~~  B  ->  ~P ( B  +c  1o )  ~~  ~P B )
6462, 63syl 17 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P ( B  +c  1o )  ~~  ~P B )
65 entr 6846 . . . . . . 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 645 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P B )
67 domentr 6853 . . . . . 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 645 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ~P B )
69 gchor 8182 . . . . 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 1188 . . . 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 372 . . 3  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  B )
7271ex 425 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  ->  ~P A  ~<_  B )
)
73 reldom 6802 . . . . 5  |-  Rel  ~<_
7473brrelexi 4682 . . . 4  |-  ( ~P A  ~<_  B  ->  ~P A  e.  _V )
75 pwexb 4501 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
76 canth2g 6948 . . . . 5  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7775, 76sylbir 206 . . . 4  |-  ( ~P A  e.  _V  ->  A 
~<  ~P A )
7874, 77syl 17 . . 3  |-  ( ~P A  ~<_  B  ->  A  ~<  ~P A )
79 sdomdomtr 6927 . . 3  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<  B )
8078, 79mpancom 653 . 2  |-  ( ~P A  ~<_  B  ->  A  ~<  B )
8172, 80impbid1 196 1  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  <->  ~P A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    e. wcel 1621   _Vcvv 2740   ~Pcpw 3566   class class class wbr 3963   omcom 4593  (class class class)co 5757   1oc1o 6405    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795   Fincfn 6796    +c ccda 7726  GCHcgch 8175
This theorem is referenced by:  gchaleph2  8231  gchina  8254
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-seqom 6393  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-oexp 6418  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-wdom 7206  df-cnf 7296  df-card 7505  df-cda 7727  df-fin4 7846  df-gch 8176
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