MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gchdomtri Unicode version

Theorem gchdomtri 8251
Description: Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8295. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchdomtri  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  ~<_  B  \/  B  ~<_  A ) )

Proof of Theorem gchdomtri
StepHypRef Expression
1 sdomdom 6889 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
21con3i 127 . . . 4  |-  ( -.  A  ~<_  B  ->  -.  A  ~<  B )
3 reldom 6869 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4729 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
543ad2ant3 978 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  e.  _V )
6 fidomtri2 7627 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  Fin )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
75, 6sylan 457 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
82, 7syl5ibr 212 . . 3  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
98orrd 367 . 2  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
10 simp1 955 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  A  e. GCH )
1110adantr 451 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  A  e. GCH )
12 simpr 447 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
13 cdadom3 7814 . . . . . 6  |-  ( ( A  e. GCH  /\  B  e.  _V )  ->  A  ~<_  ( A  +c  B
) )
1410, 5, 13syl2anc 642 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  A  ~<_  ( A  +c  B ) )
1514adantr 451 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  B ) )
16 cdalepw 7822 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
17163adant1 973 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  +c  B )  ~<_  ~P A
)
1817adantr 451 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  +c  B )  ~<_  ~P A
)
19 gchor 8249 . . . 4  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  +c  B )  /\  ( A  +c  B )  ~<_  ~P A
) )  ->  ( A  ~~  ( A  +c  B )  \/  ( A  +c  B )  ~~  ~P A ) )
2011, 12, 15, 18, 19syl22anc 1183 . . 3  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  ~~  ( A  +c  B
)  \/  ( A  +c  B )  ~~  ~P A ) )
21 cdadom3 7814 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  A  e. GCH )  ->  B  ~<_  ( B  +c  A
) )
225, 10, 21syl2anc 642 . . . . . . . 8  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  ~<_  ( B  +c  A ) )
23 cdacomen 7807 . . . . . . . 8  |-  ( B  +c  A )  ~~  ( A  +c  B
)
24 domentr 6920 . . . . . . . 8  |-  ( ( B  ~<_  ( B  +c  A )  /\  ( B  +c  A )  ~~  ( A  +c  B
) )  ->  B  ~<_  ( A  +c  B
) )
2522, 23, 24sylancl 643 . . . . . . 7  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  ~<_  ( A  +c  B ) )
26 domen2 7004 . . . . . . 7  |-  ( A 
~~  ( A  +c  B )  ->  ( B  ~<_  A  <->  B  ~<_  ( A  +c  B ) ) )
2725, 26syl5ibrcom 213 . . . . . 6  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  ~~  ( A  +c  B
)  ->  B  ~<_  A ) )
2827imp 418 . . . . 5  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  ~~  ( A  +c  B ) )  ->  B  ~<_  A )
2928olcd 382 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  ~~  ( A  +c  B ) )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
30 simpl1 958 . . . . . . 7  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  e. GCH )
31 canth2g 7015 . . . . . . 7  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
32 sdomdom 6889 . . . . . . 7  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
3330, 31, 323syl 18 . . . . . 6  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  ~<_  ~P A
)
34 simpl2 959 . . . . . . . . 9  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  +c  A )  ~~  A
)
35 pwen 7034 . . . . . . . . 9  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
3634, 35syl 15 . . . . . . . 8  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ~P ( A  +c  A )  ~~  ~P A )
37 enen2 7002 . . . . . . . . 9  |-  ( ( A  +c  B ) 
~~  ~P A  ->  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  <->  ~P ( A  +c  A )  ~~  ~P A ) )
3837adantl 452 . . . . . . . 8  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  <->  ~P ( A  +c  A )  ~~  ~P A ) )
3936, 38mpbird 223 . . . . . . 7  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ~P ( A  +c  A )  ~~  ( A  +c  B
) )
40 endom 6888 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~<_  ( A  +c  B ) )
41 pwcdadom 7842 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
4239, 40, 413syl 18 . . . . . 6  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ~P A  ~<_  B )
43 domtr 6914 . . . . . 6  |-  ( ( A  ~<_  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<_  B )
4433, 42, 43syl2anc 642 . . . . 5  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  ~<_  B )
4544orcd 381 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
4629, 45jaodan 760 . . 3  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  ~~  ( A  +c  B )  \/  ( A  +c  B
)  ~~  ~P A
) )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
4720, 46syldan 456 . 2  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
489, 47pm2.61dan 766 1  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863    +c ccda 7793  GCHcgch 8242
This theorem is referenced by:  gchaclem  8292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-wdom 7273  df-card 7572  df-cda 7794  df-gch 8243
  Copyright terms: Public domain W3C validator