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Theorem pwcdadom 8613
Description: A property of dominance over a powerset, and a main lemma for gchac 9076. 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 8023 . . . 4  |-  -.  ~P A  ~<_*  A
2 0elpw 4625 . . . . . . . . . . 11  |-  (/)  e.  ~P ( A  +c  A
)
3 n0i 3798 . . . . . . . . . . 11  |-  ( (/)  e.  ~P ( A  +c  A )  ->  -.  ~P ( A  +c  A
)  =  (/) )
42, 3ax-mp 5 . . . . . . . . . 10  |-  -.  ~P ( A  +c  A
)  =  (/)
5 dom0 7664 . . . . . . . . . 10  |-  ( ~P ( A  +c  A
)  ~<_  (/)  <->  ~P ( A  +c  A )  =  (/) )
64, 5mtbir 299 . . . . . . . . 9  |-  -.  ~P ( A  +c  A
)  ~<_  (/)
7 cdafn 8566 . . . . . . . . . . . 12  |-  +c  Fn  ( _V  X.  _V )
8 fndm 5686 . . . . . . . . . . . 12  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
97, 8ax-mp 5 . . . . . . . . . . 11  |-  dom  +c  =  ( _V  X.  _V )
109ndmov 6458 . . . . . . . . . 10  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
1110breq2d 4468 . . . . . . . . 9  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ~P ( A  +c  A )  ~<_  ( A  +c  B )  <->  ~P ( A  +c  A
)  ~<_  (/) ) )
126, 11mtbiri 303 . . . . . . . 8  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ~P ( A  +c  A )  ~<_  ( A  +c  B ) )
1312con4i 130 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
1413simpld 459 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  A  e.  _V )
15 0ex 4587 . . . . . 6  |-  (/)  e.  _V
16 xpsneng 7621 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
1714, 15, 16sylancl 662 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  X.  { (/) } ) 
~~  A )
18 endom 7561 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( A  X.  { (/)
} )  ~<_  A )
19 domwdom 8018 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_  A  ->  ( A  X.  { (/) } )  ~<_*  A )
20 wdomtr 8019 . . . . . 6  |-  ( ( ~P A  ~<_*  ( A  X.  { (/)
} )  /\  ( A  X.  { (/) } )  ~<_*  A )  ->  ~P A  ~<_*  A )
2120expcom 435 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_*  A  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
2217, 18, 19, 214syl 21 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
231, 22mtoi 178 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  -.  ~P A  ~<_*  ( A  X.  { (/)
} ) )
24 pwcdaen 8582 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
2514, 14, 24syl2anc 661 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~~  ( ~P A  X.  ~P A ) )
26 domen1 7678 . . . . . . . 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 ) ) )
2725, 26syl 16 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  <->  ( ~P A  X.  ~P A )  ~<_  ( A  +c  B
) ) )
2827ibi 241 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( A  +c  B ) )
29 cdaval 8567 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
3013, 29syl 16 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  +c  B )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
3128, 30breqtrd 4480 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
32 unxpwdom 8033 . . . . 5  |-  ( ( ~P A  X.  ~P A )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3331, 32syl 16 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3433ord 377 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( -.  ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3523, 34mpd 15 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  ( B  X.  { 1o } ) )
3613simprd 463 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  B  e.  _V )
37 1on 7155 . . 3  |-  1o  e.  On
38 xpsneng 7621 . . 3  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
3936, 37, 38sylancl 662 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( B  X.  { 1o }
)  ~~  B )
40 domentr 7593 . 2  |-  ( ( ~P A  ~<_  ( B  X.  { 1o }
)  /\  ( B  X.  { 1o } ) 
~~  B )  ->  ~P A  ~<_  B )
4135, 39, 40syl2anc 661 1  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   Oncon0 4887    X. cxp 5006   dom cdm 5008    Fn wfn 5589  (class class class)co 6296   1oc1o 7141    ~~ cen 7532    ~<_ cdom 7533    ~<_* cwdom 8001    +c ccda 8564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-1o 7148  df-2o 7149  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-wdom 8003  df-cda 8565
This theorem is referenced by:  gchdomtri  9024  gchpwdom  9065  gchhar  9074
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