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Theorem gchdomtri 9024
Description: Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9076. (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 7562 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
21con3i 135 . . . 4  |-  ( -.  A  ~<_  B  ->  -.  A  ~<  B )
3 reldom 7541 . . . . . . 7  |-  Rel  ~<_
43brrelexi 5049 . . . . . 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 8392 . . . . 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 8585 . . . . . 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 8593 . . . . . 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 9022 . . . 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 8585 . . . . . . . . 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 8578 . . . . . . . 8  |-  ( B  +c  A )  ~~  ( A  +c  B
)
24 domentr 7593 . . . . . . . 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 7679 . . . . . . 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 7690 . . . . . . 7  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
32 sdomdom 7562 . . . . . . 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 7709 . . . . . . . . 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 7677 . . . . . . . . 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 7561 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~<_  ( A  +c  B ) )
41 pwcdadom 8613 . . . . . . 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 7587 . . . . . 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 785 . . 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 791 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 1819   _Vcvv 3109   ~Pcpw 4015   class class class wbr 4456  (class class class)co 6296    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535    +c ccda 8564  GCHcgch 9015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-2o 7149  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-wdom 8003  df-card 8337  df-cda 8565  df-gch 9016
This theorem is referenced by:  gchaclem  9073
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