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Theorem pwcdadom 7837
Description: A property of dominance over a powerset, and a main lemma for gchac 8290. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
pwcdadom  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )

Proof of Theorem pwcdadom
StepHypRef Expression
1 canthwdom 7288 . . . 4  |-  -.  ~P A  ~<_*  A
2 0elpw 4179 . . . . . . . . . . . 12  |-  (/)  e.  ~P ( A  +c  A
)
3 n0i 3461 . . . . . . . . . . . 12  |-  ( (/)  e.  ~P ( A  +c  A )  ->  -.  ~P ( A  +c  A
)  =  (/) )
42, 3ax-mp 10 . . . . . . . . . . 11  |-  -.  ~P ( A  +c  A
)  =  (/)
5 dom0 6984 . . . . . . . . . . 11  |-  ( ~P ( A  +c  A
)  ~<_  (/)  <->  ~P ( A  +c  A )  =  (/) )
64, 5mtbir 292 . . . . . . . . . 10  |-  -.  ~P ( A  +c  A
)  ~<_  (/)
7 cdafn 7790 . . . . . . . . . . . . 13  |-  +c  Fn  ( _V  X.  _V )
8 fndm 5308 . . . . . . . . . . . . 13  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
97, 8ax-mp 10 . . . . . . . . . . . 12  |-  dom  +c  =  ( _V  X.  _V )
109ndmov 5965 . . . . . . . . . . 11  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
1110breq2d 4036 . . . . . . . . . 10  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ~P ( A  +c  A )  ~<_  ( A  +c  B )  <->  ~P ( A  +c  A
)  ~<_  (/) ) )
126, 11mtbiri 296 . . . . . . . . 9  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ~P ( A  +c  A )  ~<_  ( A  +c  B ) )
1312con4i 124 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
1413simpld 447 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  A  e.  _V )
15 0ex 4151 . . . . . . 7  |-  (/)  e.  _V
16 xpsneng 6942 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
1714, 15, 16sylancl 645 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  X.  { (/) } ) 
~~  A )
18 endom 6883 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( A  X.  { (/)
} )  ~<_  A )
19 domwdom 7283 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~<_  A  ->  ( A  X.  { (/) } )  ~<_*  A )
2017, 18, 193syl 20 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  X.  { (/) } )  ~<_*  A )
21 wdomtr 7284 . . . . . 6  |-  ( ( ~P A  ~<_*  ( A  X.  { (/)
} )  /\  ( A  X.  { (/) } )  ~<_*  A )  ->  ~P A  ~<_*  A )
2221expcom 426 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_*  A  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
2320, 22syl 17 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
241, 23mtoi 171 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  -.  ~P A  ~<_*  ( A  X.  { (/)
} ) )
25 pwcdaen 7806 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
2614, 14, 25syl2anc 644 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~~  ( ~P A  X.  ~P A ) )
27 domen1 6998 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~~  ( ~P A  X.  ~P A )  ->  ( ~P ( A  +c  A )  ~<_  ( A  +c  B )  <-> 
( ~P A  X.  ~P A )  ~<_  ( A  +c  B ) ) )
2826, 27syl 17 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  <->  ( ~P A  X.  ~P A )  ~<_  ( A  +c  B
) ) )
2928ibi 234 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( A  +c  B ) )
30 cdaval 7791 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
3113, 30syl 17 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  +c  B )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
3229, 31breqtrd 4048 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
33 unxpwdom 7298 . . . . 5  |-  ( ( ~P A  X.  ~P A )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3432, 33syl 17 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3534ord 368 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( -.  ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3624, 35mpd 16 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  ( B  X.  { 1o } ) )
3713simprd 451 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  B  e.  _V )
38 1on 6481 . . 3  |-  1o  e.  On
39 xpsneng 6942 . . 3  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
4037, 38, 39sylancl 645 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( B  X.  { 1o }
)  ~~  B )
41 domentr 6915 . 2  |-  ( ( ~P A  ~<_  ( B  X.  { 1o }
)  /\  ( B  X.  { 1o } ) 
~~  B )  ->  ~P A  ~<_  B )
4236, 40, 41syl2anc 644 1  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789    u. cun 3151   (/)c0 3456   ~Pcpw 3626   {csn 3641   class class class wbr 4024   Oncon0 4391    X. cxp 4686   dom cdm 4688    Fn wfn 5216  (class class class)co 5819   1oc1o 6467    ~~ cen 6855    ~<_ cdom 6856    ~<_* cwdom 7266    +c ccda 7788
This theorem is referenced by:  gchdomtri  8246  gchhar  8288  gchpwdom  8291
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-1o 6474  df-2o 6475  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-wdom 7268  df-cda 7789
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