MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gchpwdom Structured version   Unicode version

Theorem gchpwdom 8837
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 992 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  e. GCH )
2 pwexg 4476 . . . . . . 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 993 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  e. GCH )
5 cdadom3 8357 . . . . . 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 7454 . . . . 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 8350 . . . . . . 7  |-  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B )
10 entr 7361 . . . . . . 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 7358 . . . . . 6  |-  ( ( B  +c  ~P A
)  ~~  ~P B  ->  ~P B  ~~  ( B  +c  ~P A ) )
13 endom 7336 . . . . . 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 7446 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  ~<  B )  ->  om  ~<  B )
16153ad2antl1 1150 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<  B )
17 sdomnsym 7436 . . . . . . . . . 10  |-  ( om 
~<  B  ->  -.  B  ~<  om )
1816, 17syl 16 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  ~<  om )
19 isfinite 7858 . . . . . . . . 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 8835 . . . . . . . 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 7484 . . . . . . 7  |-  ( ( B  +c  B ) 
~~  B  ->  ~P ( B  +c  B
)  ~~  ~P B
)
24 domen1 7453 . . . . . . 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 8385 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P B  ~<_  ~P A )
27 canth2g 7465 . . . . . . . . 9  |-  ( B  e. GCH  ->  B  ~<  ~P B
)
28 sdomdomtr 7444 . . . . . . . . . 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 8791 . . . . . . . . . 10  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
32313expia 1189 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
33323ad2antl2 1151 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin ) )
34 isfinite 7858 . . . . . . . . 9  |-  ( A  e.  Fin  <->  A  ~<  om )
35 simpl1 991 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<_  A )
36 domnsym 7437 . . . . . . . . . . 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 8357 . . . . . . 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 7368 . . . . . 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 7337 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  ~<_  B )
4948adantl 466 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  ~<_  B )
50 pwdom 7463 . . . . . . . 8  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
51 cdadom1 8355 . . . . . . . 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 7337 . . . . . . . 8  |-  ( B 
~<  ~P B  ->  B  ~<_  ~P B )
55 cdadom2 8356 . . . . . . . 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 7362 . . . . . . 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 8363 . . . . . . . 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 8823 . . . . . . . . 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 7484 . . . . . . . 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 7361 . . . . . . 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 7368 . . . . . 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 8794 . . . . 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 1219 . . . 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 7316 . . . . 5  |-  Rel  ~<_
7473brrelexi 4879 . . . 4  |-  ( ~P A  ~<_  B  ->  ~P A  e.  _V )
75 pwexb 6387 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
76 canth2g 7465 . . . . 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 7444 . . 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 965    e. wcel 1756   _Vcvv 2972   ~Pcpw 3860   class class class wbr 4292  (class class class)co 6091   omcom 6476   1oc1o 6913    ~~ cen 7307    ~<_ cdom 7308    ~< csdm 7309   Fincfn 7310    +c ccda 8336  GCHcgch 8787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-seqom 6903  df-1o 6920  df-2o 6921  df-oadd 6924  df-omul 6925  df-oexp 6926  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-har 7773  df-wdom 7774  df-cnf 7868  df-card 8109  df-cda 8337  df-fin4 8456  df-gch 8788
This theorem is referenced by:  gchaleph2  8839  gchina  8866
  Copyright terms: Public domain W3C validator