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Theorem gchdomtri 8796
Description: Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8848. (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 7337 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
21con3i 135 . . . 4  |-  ( -.  A  ~<_  B  ->  -.  A  ~<  B )
3 reldom 7316 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4879 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
543ad2ant3 1011 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  e.  _V )
6 fidomtri2 8164 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  Fin )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
75, 6sylan 471 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
82, 7syl5ibr 221 . . 3  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
98orrd 378 . 2  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
10 simp1 988 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  A  e. GCH )
1110adantr 465 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  A  e. GCH )
12 simpr 461 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
13 cdadom3 8357 . . . . . 6  |-  ( ( A  e. GCH  /\  B  e.  _V )  ->  A  ~<_  ( A  +c  B
) )
1410, 5, 13syl2anc 661 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  A  ~<_  ( A  +c  B ) )
1514adantr 465 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  B ) )
16 cdalepw 8365 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
17163adant1 1006 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  +c  B )  ~<_  ~P A
)
1817adantr 465 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  +c  B )  ~<_  ~P A
)
19 gchor 8794 . . . 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 1219 . . 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 8357 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  A  e. GCH )  ->  B  ~<_  ( B  +c  A
) )
225, 10, 21syl2anc 661 . . . . . . . 8  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  ~<_  ( B  +c  A ) )
23 cdacomen 8350 . . . . . . . 8  |-  ( B  +c  A )  ~~  ( A  +c  B
)
24 domentr 7368 . . . . . . . 8  |-  ( ( B  ~<_  ( B  +c  A )  /\  ( B  +c  A )  ~~  ( A  +c  B
) )  ->  B  ~<_  ( A  +c  B
) )
2522, 23, 24sylancl 662 . . . . . . 7  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  ~<_  ( A  +c  B ) )
26 domen2 7454 . . . . . . 7  |-  ( A 
~~  ( A  +c  B )  ->  ( B  ~<_  A  <->  B  ~<_  ( A  +c  B ) ) )
2725, 26syl5ibrcom 222 . . . . . 6  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  ~~  ( A  +c  B
)  ->  B  ~<_  A ) )
2827imp 429 . . . . 5  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  ~~  ( A  +c  B ) )  ->  B  ~<_  A )
2928olcd 393 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  ~~  ( A  +c  B ) )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
30 simpl1 991 . . . . . . 7  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  e. GCH )
31 canth2g 7465 . . . . . . 7  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
32 sdomdom 7337 . . . . . . 7  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
3330, 31, 323syl 20 . . . . . 6  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  ~<_  ~P A
)
34 simpl2 992 . . . . . . . . 9  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  +c  A )  ~~  A
)
35 pwen 7484 . . . . . . . . 9  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
3634, 35syl 16 . . . . . . . 8  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ~P ( A  +c  A )  ~~  ~P A )
37 enen2 7452 . . . . . . . . 9  |-  ( ( A  +c  B ) 
~~  ~P A  ->  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  <->  ~P ( A  +c  A )  ~~  ~P A ) )
3837adantl 466 . . . . . . . 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 232 . . . . . . 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 7336 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~<_  ( A  +c  B ) )
41 pwcdadom 8385 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
4239, 40, 413syl 20 . . . . . 6  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ~P A  ~<_  B )
43 domtr 7362 . . . . . 6  |-  ( ( A  ~<_  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<_  B )
4433, 42, 43syl2anc 661 . . . . 5  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  ~<_  B )
4544orcd 392 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
4629, 45jaodan 783 . . 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 470 . 2  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
489, 47pm2.61dan 789 1  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    e. wcel 1756   _Vcvv 2972   ~Pcpw 3860   class class class wbr 4292  (class class class)co 6091    ~~ cen 7307    ~<_ cdom 7308    ~< csdm 7309   Fincfn 7310    +c ccda 8336  GCHcgch 8787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-1o 6920  df-2o 6921  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-wdom 7774  df-card 8109  df-cda 8337  df-gch 8788
This theorem is referenced by:  gchaclem  8845
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