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Theorem pwcdadom 8489
Description: A property of dominance over a powerset, and a main lemma for gchac 8952. 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 7898 . . . 4  |-  -.  ~P A  ~<_*  A
2 0elpw 4562 . . . . . . . . . . 11  |-  (/)  e.  ~P ( A  +c  A
)
3 n0i 3743 . . . . . . . . . . 11  |-  ( (/)  e.  ~P ( A  +c  A )  ->  -.  ~P ( A  +c  A
)  =  (/) )
42, 3ax-mp 5 . . . . . . . . . 10  |-  -.  ~P ( A  +c  A
)  =  (/)
5 dom0 7542 . . . . . . . . . 10  |-  ( ~P ( A  +c  A
)  ~<_  (/)  <->  ~P ( A  +c  A )  =  (/) )
64, 5mtbir 299 . . . . . . . . 9  |-  -.  ~P ( A  +c  A
)  ~<_  (/)
7 cdafn 8442 . . . . . . . . . . . 12  |-  +c  Fn  ( _V  X.  _V )
8 fndm 5611 . . . . . . . . . . . 12  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
97, 8ax-mp 5 . . . . . . . . . . 11  |-  dom  +c  =  ( _V  X.  _V )
109ndmov 6350 . . . . . . . . . 10  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
1110breq2d 4405 . . . . . . . . 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 4523 . . . . . 6  |-  (/)  e.  _V
16 xpsneng 7499 . . . . . 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 7439 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( A  X.  { (/)
} )  ~<_  A )
19 domwdom 7893 . . . . 5  |-  ( ( A  X.  { (/) } )  ~<_  A  ->  ( A  X.  { (/) } )  ~<_*  A )
20 wdomtr 7894 . . . . . 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 8458 . . . . . . . . 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 7556 . . . . . . . 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 8443 . . . . . . 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 4417 . . . . 5  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ( ~P A  X.  ~P A
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
32 unxpwdom 7908 . . . . 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 7030 . . 3  |-  1o  e.  On
38 xpsneng 7499 . . 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 7471 . 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 1370    e. wcel 1758   _Vcvv 3071    u. cun 3427   (/)c0 3738   ~Pcpw 3961   {csn 3978   class class class wbr 4393   Oncon0 4820    X. cxp 4939   dom cdm 4941    Fn wfn 5514  (class class class)co 6193   1oc1o 7016    ~~ cen 7410    ~<_ cdom 7411    ~<_* cwdom 7876    +c ccda 8440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-1o 7023  df-2o 7024  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-wdom 7878  df-cda 8441
This theorem is referenced by:  gchdomtri  8900  gchpwdom  8941  gchhar  8950
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