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Theorem pwcdadom 8052
Description: A property of dominance over a powerset, and a main lemma for gchac 8504. 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 7503 . . . 4  |-  -.  ~P A  ~<_*  A
2 0elpw 4329 . . . . . . . . . . . 12  |-  (/)  e.  ~P ( A  +c  A
)
3 n0i 3593 . . . . . . . . . . . 12  |-  ( (/)  e.  ~P ( A  +c  A )  ->  -.  ~P ( A  +c  A
)  =  (/) )
42, 3ax-mp 8 . . . . . . . . . . 11  |-  -.  ~P ( A  +c  A
)  =  (/)
5 dom0 7194 . . . . . . . . . . 11  |-  ( ~P ( A  +c  A
)  ~<_  (/)  <->  ~P ( A  +c  A )  =  (/) )
64, 5mtbir 291 . . . . . . . . . 10  |-  -.  ~P ( A  +c  A
)  ~<_  (/)
7 cdafn 8005 . . . . . . . . . . . . 13  |-  +c  Fn  ( _V  X.  _V )
8 fndm 5503 . . . . . . . . . . . . 13  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
97, 8ax-mp 8 . . . . . . . . . . . 12  |-  dom  +c  =  ( _V  X.  _V )
109ndmov 6190 . . . . . . . . . . 11  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
1110breq2d 4184 . . . . . . . . . 10  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ~P ( A  +c  A )  ~<_  ( A  +c  B )  <->  ~P ( A  +c  A
)  ~<_  (/) ) )
126, 11mtbiri 295 . . . . . . . . 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 446 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  A  e.  _V )
15 0ex 4299 . . . . . . 7  |-  (/)  e.  _V
16 xpsneng 7152 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
1714, 15, 16sylancl 644 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  X.  { (/) } ) 
~~  A )
18 endom 7093 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( A  X.  { (/)
} )  ~<_  A )
19 domwdom 7498 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~<_  A  ->  ( A  X.  { (/) } )  ~<_*  A )
2017, 18, 193syl 19 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  X.  { (/) } )  ~<_*  A )
21 wdomtr 7499 . . . . . 6  |-  ( ( ~P A  ~<_*  ( A  X.  { (/)
} )  /\  ( A  X.  { (/) } )  ~<_*  A )  ->  ~P A  ~<_*  A )
2221expcom 425 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_*  A  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
2320, 22syl 16 . . . 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 8021 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
2614, 14, 25syl2anc 643 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~~  ( ~P A  X.  ~P A ) )
27 domen1 7208 . . . . . . . 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 16 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  <->  ( ~P A  X.  ~P A )  ~<_  ( A  +c  B
) ) )
2928ibi 233 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( A  +c  B ) )
30 cdaval 8006 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
3113, 30syl 16 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  +c  B )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
3229, 31breqtrd 4196 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
33 unxpwdom 7513 . . . . 5  |-  ( ( ~P A  X.  ~P A )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3432, 33syl 16 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3534ord 367 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( -.  ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3624, 35mpd 15 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  ( B  X.  { 1o } ) )
3713simprd 450 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  B  e.  _V )
38 1on 6690 . . 3  |-  1o  e.  On
39 xpsneng 7152 . . 3  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
4037, 38, 39sylancl 644 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( B  X.  { 1o }
)  ~~  B )
41 domentr 7125 . 2  |-  ( ( ~P A  ~<_  ( B  X.  { 1o }
)  /\  ( B  X.  { 1o } ) 
~~  B )  ->  ~P A  ~<_  B )
4236, 40, 41syl2anc 643 1  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   (/)c0 3588   ~Pcpw 3759   {csn 3774   class class class wbr 4172   Oncon0 4541    X. cxp 4835   dom cdm 4837    Fn wfn 5408  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    ~<_* cwdom 7481    +c ccda 8003
This theorem is referenced by:  gchdomtri  8460  gchhar  8502  gchpwdom  8505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-2o 6684  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-wdom 7483  df-cda 8004
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