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Theorem gchdomtri 9003
Description: Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9055. (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 7540 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
21con3i 135 . . . 4  |-  ( -.  A  ~<_  B  ->  -.  A  ~<  B )
3 reldom 7519 . . . . . . 7  |-  Rel  ~<_
43brrelexi 5039 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
543ad2ant3 1019 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  e.  _V )
6 fidomtri2 8371 . . . . 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 996 . . . . 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 8564 . . . . . 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 8572 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
17163adant1 1014 . . . . 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 9001 . . . 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 1229 . . 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 8564 . . . . . . . . 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 8557 . . . . . . . 8  |-  ( B  +c  A )  ~~  ( A  +c  B
)
24 domentr 7571 . . . . . . . 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 7657 . . . . . . 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 999 . . . . . . 7  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  e. GCH )
31 canth2g 7668 . . . . . . 7  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
32 sdomdom 7540 . . . . . . 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 1000 . . . . . . . . 9  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  +c  A )  ~~  A
)
35 pwen 7687 . . . . . . . . 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 7655 . . . . . . . . 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 7539 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~<_  ( A  +c  B ) )
41 pwcdadom 8592 . . . . . . 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 7565 . . . . . 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 973    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447  (class class class)co 6282    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512   Fincfn 7513    +c ccda 8543  GCHcgch 8994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-2o 7128  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316  df-cda 8544  df-gch 8995
This theorem is referenced by:  gchaclem  9052
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