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Theorem gchdomtri 8438
Description: Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8482. (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 7072 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
21con3i 129 . . . 4  |-  ( -.  A  ~<_  B  ->  -.  A  ~<  B )
3 reldom 7052 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4859 . . . . . 6  |-  ( B  ~<_  ~P A  ->  B  e.  _V )
543ad2ant3 980 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  e.  _V )
6 fidomtri2 7815 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  Fin )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
75, 6sylan 458 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
82, 7syl5ibr 213 . . 3  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( -.  A  ~<_  B  ->  B  ~<_  A ) )
98orrd 368 . 2  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
10 simp1 957 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  A  e. GCH )
1110adantr 452 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  A  e. GCH )
12 simpr 448 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
13 cdadom3 8002 . . . . . 6  |-  ( ( A  e. GCH  /\  B  e.  _V )  ->  A  ~<_  ( A  +c  B
) )
1410, 5, 13syl2anc 643 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  A  ~<_  ( A  +c  B ) )
1514adantr 452 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  B ) )
16 cdalepw 8010 . . . . . 6  |-  ( ( ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  -> 
( A  +c  B
)  ~<_  ~P A )
17163adant1 975 . . . . 5  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  +c  B )  ~<_  ~P A
)
1817adantr 452 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  +c  B )  ~<_  ~P A
)
19 gchor 8436 . . . 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 1185 . . 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 8002 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  A  e. GCH )  ->  B  ~<_  ( B  +c  A
) )
225, 10, 21syl2anc 643 . . . . . . . 8  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  ~<_  ( B  +c  A ) )
23 cdacomen 7995 . . . . . . . 8  |-  ( B  +c  A )  ~~  ( A  +c  B
)
24 domentr 7103 . . . . . . . 8  |-  ( ( B  ~<_  ( B  +c  A )  /\  ( B  +c  A )  ~~  ( A  +c  B
) )  ->  B  ~<_  ( A  +c  B
) )
2522, 23, 24sylancl 644 . . . . . . 7  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  B  ~<_  ( A  +c  B ) )
26 domen2 7187 . . . . . . 7  |-  ( A 
~~  ( A  +c  B )  ->  ( B  ~<_  A  <->  B  ~<_  ( A  +c  B ) ) )
2725, 26syl5ibrcom 214 . . . . . 6  |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A
)  ->  ( A  ~~  ( A  +c  B
)  ->  B  ~<_  A ) )
2827imp 419 . . . . 5  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  ~~  ( A  +c  B ) )  ->  B  ~<_  A )
2928olcd 383 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  A  ~~  ( A  +c  B ) )  -> 
( A  ~<_  B  \/  B  ~<_  A ) )
30 simpl1 960 . . . . . . 7  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  e. GCH )
31 canth2g 7198 . . . . . . 7  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
32 sdomdom 7072 . . . . . . 7  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
3330, 31, 323syl 19 . . . . . 6  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  ~<_  ~P A
)
34 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  +c  A )  ~~  A
)
35 pwen 7217 . . . . . . . . 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 7185 . . . . . . . . 9  |-  ( ( A  +c  B ) 
~~  ~P A  ->  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  <->  ~P ( A  +c  A )  ~~  ~P A ) )
3837adantl 453 . . . . . . . 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 224 . . . . . . 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 7071 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  B )  ->  ~P ( A  +c  A
)  ~<_  ( A  +c  B ) )
41 pwcdadom 8030 . . . . . . 7  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  B )  ->  ~P A  ~<_  B )
4239, 40, 413syl 19 . . . . . 6  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ~P A  ~<_  B )
43 domtr 7097 . . . . . 6  |-  ( ( A  ~<_  ~P A  /\  ~P A  ~<_  B )  ->  A  ~<_  B )
4433, 42, 43syl2anc 643 . . . . 5  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  A  ~<_  B )
4544orcd 382 . . . 4  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  ( A  +c  B
)  ~~  ~P A
)  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
4629, 45jaodan 761 . . 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 457 . 2  |-  ( ( ( A  e. GCH  /\  ( A  +c  A
)  ~~  A  /\  B  ~<_  ~P A )  /\  -.  A  e.  Fin )  ->  ( A  ~<_  B  \/  B  ~<_  A ) )
489, 47pm2.61dan 767 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1717   _Vcvv 2900   ~Pcpw 3743   class class class wbr 4154  (class class class)co 6021    ~~ cen 7043    ~<_ cdom 7044    ~< csdm 7045   Fincfn 7046    +c ccda 7981  GCHcgch 8429
This theorem is referenced by:  gchaclem  8479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-1o 6661  df-2o 6662  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-wdom 7461  df-card 7760  df-cda 7982  df-gch 8430
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