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Theorem gchpwdom 9048
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 1000 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  e. GCH )
2 pwexg 4631 . . . . . . 7  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
31, 2syl 16 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  e.  _V )
4 simpl3 1001 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  e. GCH )
5 cdadom3 8568 . . . . . 6  |-  ( ( ~P A  e.  _V  /\  B  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  B ) )
63, 4, 5syl2anc 661 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  ( ~P A  +c  B
) )
7 domen2 7660 . . . . 5  |-  ( B 
~~  ( ~P A  +c  B )  ->  ( ~P A  ~<_  B  <->  ~P A  ~<_  ( ~P A  +c  B
) ) )
86, 7syl5ibrcom 222 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~~  ( ~P A  +c  B )  ->  ~P A  ~<_  B ) )
9 cdacomen 8561 . . . . . . 7  |-  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B )
10 entr 7567 . . . . . . 7  |-  ( ( ( B  +c  ~P A )  ~~  ( ~P A  +c  B
)  /\  ( ~P A  +c  B )  ~~  ~P B )  ->  ( B  +c  ~P A ) 
~~  ~P B )
119, 10mpan 670 . . . . . 6  |-  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ( B  +c  ~P A )  ~~  ~P B )
12 ensym 7564 . . . . . 6  |-  ( ( B  +c  ~P A
)  ~~  ~P B  ->  ~P B  ~~  ( B  +c  ~P A ) )
13 endom 7542 . . . . . 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 7652 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  ~<  B )  ->  om  ~<  B )
16153ad2antl1 1158 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<  B )
17 sdomnsym 7642 . . . . . . . . . 10  |-  ( om 
~<  B  ->  -.  B  ~<  om )
1816, 17syl 16 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  ~<  om )
19 isfinite 8069 . . . . . . . . 9  |-  ( B  e.  Fin  <->  B  ~<  om )
2018, 19sylnibr 305 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  e.  Fin )
21 gchcdaidm 9046 . . . . . . . 8  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  B
)  ~~  B )
224, 20, 21syl2anc 661 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  B )  ~~  B
)
23 pwen 7690 . . . . . . 7  |-  ( ( B  +c  B ) 
~~  B  ->  ~P ( B  +c  B
)  ~~  ~P B
)
24 domen1 7659 . . . . . . 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 8596 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P B  ~<_  ~P A )
27 canth2g 7671 . . . . . . . . 9  |-  ( B  e. GCH  ->  B  ~<  ~P B
)
28 sdomdomtr 7650 . . . . . . . . . 10  |-  ( ( B  ~<  ~P B  /\  ~P B  ~<_  ~P A
)  ->  B  ~<  ~P A )
2928ex 434 . . . . . . . . 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 9002 . . . . . . . . . 10  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
32313expia 1198 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
33323ad2antl2 1159 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin ) )
34 isfinite 8069 . . . . . . . . 9  |-  ( A  e.  Fin  <->  A  ~<  om )
35 simpl1 999 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<_  A )
36 domnsym 7643 . . . . . . . . . . 11  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
3735, 36syl 16 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  A  ~<  om )
3837pm2.21d 106 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  ~<  om  ->  ~P A  ~<_  B ) )
3934, 38syl5bi 217 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  e.  Fin  ->  ~P A  ~<_  B ) )
4030, 33, 393syld 55 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ~P A  ->  ~P A  ~<_  B ) )
4126, 40syl5 32 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P A  ~<_  B ) )
4225, 41sylbird 235 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  ~<_  ( B  +c  ~P A )  ->  ~P A  ~<_  B ) )
4314, 42syl5 32 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ~P A  ~<_  B ) )
44 cdadom3 8568 . . . . . . 7  |-  ( ( B  e. GCH  /\  ~P A  e.  _V )  ->  B  ~<_  ( B  +c  ~P A ) )
454, 3, 44syl2anc 661 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( B  +c  ~P A ) )
46 domentr 7574 . . . . . 6  |-  ( ( B  ~<_  ( B  +c  ~P A )  /\  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B ) )  ->  B  ~<_  ( ~P A  +c  B ) )
4745, 9, 46sylancl 662 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( ~P A  +c  B ) )
48 sdomdom 7543 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  ~<_  B )
4948adantl 466 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  ~<_  B )
50 pwdom 7669 . . . . . . . 8  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
51 cdadom1 8566 . . . . . . . 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 16 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<  ~P B )
54 sdomdom 7543 . . . . . . . 8  |-  ( B 
~<  ~P B  ->  B  ~<_  ~P B )
55 cdadom2 8567 . . . . . . . 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 7568 . . . . . . 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 661 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ( ~P B  +c  ~P B ) )
59 pwcda1 8574 . . . . . . . 8  |-  ( B  e. GCH  ->  ( ~P B  +c  ~P B )  ~~  ~P ( B  +c  1o ) )
604, 59syl 16 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P ( B  +c  1o ) )
61 gchcda1 9034 . . . . . . . . 9  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  1o )  ~~  B )
624, 20, 61syl2anc 661 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  1o )  ~~  B
)
63 pwen 7690 . . . . . . . 8  |-  ( ( B  +c  1o ) 
~~  B  ->  ~P ( B  +c  1o )  ~~  ~P B )
6462, 63syl 16 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P ( B  +c  1o )  ~~  ~P B )
65 entr 7567 . . . . . . 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 661 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P B )
67 domentr 7574 . . . . . 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 661 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ~P B )
69 gchor 9005 . . . . 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 1229 . . . 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 381 . . 3  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  B )
7271ex 434 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  ->  ~P A  ~<_  B )
)
73 reldom 7522 . . . . 5  |-  Rel  ~<_
7473brrelexi 5040 . . . 4  |-  ( ~P A  ~<_  B  ->  ~P A  e.  _V )
75 pwexb 6595 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
76 canth2g 7671 . . . . 5  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7775, 76sylbir 213 . . . 4  |-  ( ~P A  e.  _V  ->  A 
~<  ~P A )
7874, 77syl 16 . . 3  |-  ( ~P A  ~<_  B  ->  A  ~<  ~P A )
79 sdomdomtr 7650 . . 3  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<  B )
8078, 79mpancom 669 . 2  |-  ( ~P A  ~<_  B  ->  A  ~<  B )
8172, 80impbid1 203 1  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  <->  ~P A  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447  (class class class)co 6284   omcom 6684   1oc1o 7123    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515   Fincfn 7516    +c ccda 8547  GCHcgch 8998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-seqom 7113  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-oexp 7136  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-oi 7935  df-har 7984  df-wdom 7985  df-cnf 8079  df-card 8320  df-cda 8548  df-fin4 8667  df-gch 8999
This theorem is referenced by:  gchaleph2  9050  gchina  9077
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