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Theorem gchpwdom 9113
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 1034 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  e. GCH )
2 pwexg 4585 . . . . . . 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 1035 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  e. GCH )
5 cdadom3 8636 . . . . . 6  |-  ( ( ~P A  e.  _V  /\  B  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  B ) )
63, 4, 5syl2anc 673 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  ( ~P A  +c  B
) )
7 domen2 7733 . . . . 5  |-  ( B 
~~  ( ~P A  +c  B )  ->  ( ~P A  ~<_  B  <->  ~P A  ~<_  ( ~P A  +c  B
) ) )
86, 7syl5ibrcom 230 . . . 4  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~~  ( ~P A  +c  B )  ->  ~P A  ~<_  B ) )
9 cdacomen 8629 . . . . . . 7  |-  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B )
10 entr 7639 . . . . . . 7  |-  ( ( ( B  +c  ~P A )  ~~  ( ~P A  +c  B
)  /\  ( ~P A  +c  B )  ~~  ~P B )  ->  ( B  +c  ~P A ) 
~~  ~P B )
119, 10mpan 684 . . . . . 6  |-  ( ( ~P A  +c  B
)  ~~  ~P B  ->  ( B  +c  ~P A )  ~~  ~P B )
12 ensym 7636 . . . . . 6  |-  ( ( B  +c  ~P A
)  ~~  ~P B  ->  ~P B  ~~  ( B  +c  ~P A ) )
13 endom 7614 . . . . . 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 7725 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  ~<  B )  ->  om  ~<  B )
16153ad2antl1 1192 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<  B )
17 sdomnsym 7715 . . . . . . . . . 10  |-  ( om 
~<  B  ->  -.  B  ~<  om )
1816, 17syl 17 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  ~<  om )
19 isfinite 8175 . . . . . . . . 9  |-  ( B  e.  Fin  <->  B  ~<  om )
2018, 19sylnibr 312 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  B  e.  Fin )
21 gchcdaidm 9111 . . . . . . . 8  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  B
)  ~~  B )
224, 20, 21syl2anc 673 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  B )  ~~  B
)
23 pwen 7763 . . . . . . 7  |-  ( ( B  +c  B ) 
~~  B  ->  ~P ( B  +c  B
)  ~~  ~P B
)
24 domen1 7732 . . . . . . 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 8664 . . . . . . 7  |-  ( ~P ( B  +c  B
)  ~<_  ( B  +c  ~P A )  ->  ~P B  ~<_  ~P A )
27 canth2g 7744 . . . . . . . . 9  |-  ( B  e. GCH  ->  B  ~<  ~P B
)
28 sdomdomtr 7723 . . . . . . . . . 10  |-  ( ( B  ~<  ~P B  /\  ~P B  ~<_  ~P A
)  ->  B  ~<  ~P A )
2928ex 441 . . . . . . . . 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 9067 . . . . . . . . . 10  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
32313expia 1233 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
33323ad2antl2 1193 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin ) )
34 isfinite 8175 . . . . . . . . 9  |-  ( A  e.  Fin  <->  A  ~<  om )
35 simpl1 1033 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  om  ~<_  A )
36 domnsym 7716 . . . . . . . . . . 11  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
3735, 36syl 17 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  -.  A  ~<  om )
3837pm2.21d 109 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  ~<  om  ->  ~P A  ~<_  B ) )
3934, 38syl5bi 225 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( A  e.  Fin  ->  ~P A  ~<_  B ) )
4030, 33, 393syld 56 . . . . . . 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 243 . . . . 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 8636 . . . . . . 7  |-  ( ( B  e. GCH  /\  ~P A  e.  _V )  ->  B  ~<_  ( B  +c  ~P A ) )
454, 3, 44syl2anc 673 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( B  +c  ~P A ) )
46 domentr 7646 . . . . . 6  |-  ( ( B  ~<_  ( B  +c  ~P A )  /\  ( B  +c  ~P A ) 
~~  ( ~P A  +c  B ) )  ->  B  ~<_  ( ~P A  +c  B ) )
4745, 9, 46sylancl 675 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<_  ( ~P A  +c  B ) )
48 sdomdom 7615 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  ~<_  B )
4948adantl 473 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  A  ~<_  B )
50 pwdom 7742 . . . . . . . 8  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
51 cdadom1 8634 . . . . . . . 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 17 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  B  ~<  ~P B )
54 sdomdom 7615 . . . . . . . 8  |-  ( B 
~<  ~P B  ->  B  ~<_  ~P B )
55 cdadom2 8635 . . . . . . . 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 7640 . . . . . . 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 673 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ( ~P B  +c  ~P B ) )
59 pwcda1 8642 . . . . . . . 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 9099 . . . . . . . . 9  |-  ( ( B  e. GCH  /\  -.  B  e.  Fin )  ->  ( B  +c  1o )  ~~  B )
624, 20, 61syl2anc 673 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( B  +c  1o )  ~~  B
)
63 pwen 7763 . . . . . . . 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 7639 . . . . . . 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 673 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P B  +c  ~P B ) 
~~  ~P B )
67 domentr 7646 . . . . . 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 673 . . . . 5  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ( ~P A  +c  B )  ~<_  ~P B )
69 gchor 9070 . . . . 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 1293 . . . 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 388 . . 3  |-  ( ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  /\  A  ~<  B )  ->  ~P A  ~<_  B )
7271ex 441 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH )  ->  ( A  ~<  B  ->  ~P A  ~<_  B )
)
73 reldom 7593 . . . . 5  |-  Rel  ~<_
7473brrelexi 4880 . . . 4  |-  ( ~P A  ~<_  B  ->  ~P A  e.  _V )
75 pwexb 6621 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
76 canth2g 7744 . . . . 5  |-  ( A  e.  _V  ->  A  ~<  ~P A )
7775, 76sylbir 218 . . . 4  |-  ( ~P A  e.  _V  ->  A 
~<  ~P A )
7874, 77syl 17 . . 3  |-  ( ~P A  ~<_  B  ->  A  ~<  ~P A )
79 sdomdomtr 7723 . . 3  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<  B )
8078, 79mpancom 682 . 2  |-  ( ~P A  ~<_  B  ->  A  ~<  B )
8172, 80impbid1 208 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 189    \/ wo 375    /\ wa 376    /\ w3a 1007    e. wcel 1904   _Vcvv 3031   ~Pcpw 3942   class class class wbr 4395  (class class class)co 6308   omcom 6711   1oc1o 7193    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587    +c ccda 8615  GCHcgch 9063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seqom 7183  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-oexp 7206  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-har 8091  df-wdom 8092  df-cnf 8185  df-card 8391  df-cda 8616  df-fin4 8735  df-gch 9064
This theorem is referenced by:  gchaleph2  9115  gchina  9142
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