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Theorem pwcdadom 8597
Description: A property of dominance over a powerset, and a main lemma for gchac 9057. 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 8047 . . . 4  |-  -.  ~P A  ~<_*  A
2 0elpw 4536 . . . . . . . . . . 11  |-  (/)  e.  ~P ( A  +c  A
)
3 n0i 3709 . . . . . . . . . . 11  |-  ( (/)  e.  ~P ( A  +c  A )  ->  -.  ~P ( A  +c  A
)  =  (/) )
42, 3ax-mp 5 . . . . . . . . . 10  |-  -.  ~P ( A  +c  A
)  =  (/)
5 dom0 7653 . . . . . . . . . 10  |-  ( ~P ( A  +c  A
)  ~<_  (/)  <->  ~P ( A  +c  A )  =  (/) )
64, 5mtbir 300 . . . . . . . . 9  |-  -.  ~P ( A  +c  A
)  ~<_  (/)
7 cdafn 8550 . . . . . . . . . . . 12  |-  +c  Fn  ( _V  X.  _V )
8 fndm 5636 . . . . . . . . . . . 12  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
97, 8ax-mp 5 . . . . . . . . . . 11  |-  dom  +c  =  ( _V  X.  _V )
109ndmov 6411 . . . . . . . . . 10  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
1110breq2d 4378 . . . . . . . . 9  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ~P ( A  +c  A )  ~<_  ( A  +c  B )  <->  ~P ( A  +c  A
)  ~<_  (/) ) )
126, 11mtbiri 304 . . . . . . . 8  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ~P ( A  +c  A )  ~<_  ( A  +c  B ) )
1312con4i 133 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
1413simpld 460 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  A  e.  _V )
15 0ex 4499 . . . . . 6  |-  (/)  e.  _V
16 xpsneng 7610 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
1714, 15, 16sylancl 666 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  X.  { (/) } ) 
~~  A )
18 endom 7550 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( A  X.  { (/)
} )  ~<_  A )
19 domwdom 8042 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_  A  ->  ( A  X.  { (/) } )  ~<_*  A )
20 wdomtr 8043 . . . . . 6  |-  ( ( ~P A  ~<_*  ( A  X.  { (/)
} )  /\  ( A  X.  { (/) } )  ~<_*  A )  ->  ~P A  ~<_*  A )
2120expcom 436 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_*  A  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
2217, 18, 19, 214syl 19 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  ->  ~P A  ~<_*  A ) )
231, 22mtoi 181 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  -.  ~P A  ~<_*  ( A  X.  { (/)
} ) )
24 pwcdaen 8566 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
2514, 14, 24syl2anc 665 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~~  ( ~P A  X.  ~P A ) )
26 domen1 7667 . . . . . . . 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 17 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  <->  ( ~P A  X.  ~P A )  ~<_  ( A  +c  B
) ) )
2827ibi 244 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( A  +c  B ) )
29 cdaval 8551 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
3013, 29syl 17 . . . . . 6  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( A  +c  B )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
3128, 30breqtrd 4391 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
32 unxpwdom 8057 . . . . 5  |-  ( ( ~P A  X.  ~P A )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3331, 32syl 17 . . . 4  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  ~<_*  ( A  X.  { (/)
} )  \/  ~P A  ~<_  ( B  X.  { 1o } ) ) )
3433ord 378 . . 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 464 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  B  e.  _V )
37 1on 7144 . . 3  |-  1o  e.  On
38 xpsneng 7610 . . 3  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
3936, 37, 38sylancl 666 . 2  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( B  X.  { 1o }
)  ~~  B )
40 domentr 7582 . 2  |-  ( ( ~P A  ~<_  ( B  X.  { 1o }
)  /\  ( B  X.  { 1o } ) 
~~  B )  ->  ~P A  ~<_  B )
4135, 39, 40syl2anc 665 1  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    u. cun 3377   (/)c0 3704   ~Pcpw 3924   {csn 3941   class class class wbr 4366    X. cxp 4794   dom cdm 4796   Oncon0 5385    Fn wfn 5539  (class class class)co 6249   1oc1o 7130    ~~ cen 7521    ~<_ cdom 7522    ~<_* cwdom 8025    +c ccda 8548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-1o 7137  df-2o 7138  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-wdom 8027  df-cda 8549
This theorem is referenced by:  gchdomtri  9005  gchpwdom  9046  gchhar  9055
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