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Theorem gchaclem 9052
Description: Lemma for gchac 9055 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
gchaclem.1  |-  ( ph  ->  om  ~<_  A )
gchaclem.3  |-  ( ph  ->  ~P C  e. GCH )
gchaclem.4  |-  ( ph  ->  ( A  ~<_  C  /\  ( B  ~<_  ~P C  ->  ~P A  ~<_  B ) ) )
Assertion
Ref Expression
gchaclem  |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )

Proof of Theorem gchaclem
StepHypRef Expression
1 gchaclem.4 . . . 4  |-  ( ph  ->  ( A  ~<_  C  /\  ( B  ~<_  ~P C  ->  ~P A  ~<_  B ) ) )
21simpld 459 . . 3  |-  ( ph  ->  A  ~<_  C )
3 reldom 7519 . . . . . 6  |-  Rel  ~<_
43brrelex2i 5040 . . . . 5  |-  ( A  ~<_  C  ->  C  e.  _V )
52, 4syl 16 . . . 4  |-  ( ph  ->  C  e.  _V )
6 canth2g 7668 . . . 4  |-  ( C  e.  _V  ->  C  ~<  ~P C )
7 sdomdom 7540 . . . 4  |-  ( C 
~<  ~P C  ->  C  ~<_  ~P C )
85, 6, 73syl 20 . . 3  |-  ( ph  ->  C  ~<_  ~P C )
9 domtr 7565 . . 3  |-  ( ( A  ~<_  C  /\  C  ~<_  ~P C )  ->  A  ~<_  ~P C )
102, 8, 9syl2anc 661 . 2  |-  ( ph  ->  A  ~<_  ~P C )
11 gchaclem.3 . . . . . 6  |-  ( ph  ->  ~P C  e. GCH )
1211adantr 465 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ~P C  e. GCH )
13 gchaclem.1 . . . . . . . 8  |-  ( ph  ->  om  ~<_  A )
14 domtr 7565 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  C )  ->  om  ~<_  C )
1513, 2, 14syl2anc 661 . . . . . . 7  |-  ( ph  ->  om  ~<_  C )
1615adantr 465 . . . . . 6  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  om  ~<_  C )
17 pwcdaidm 8571 . . . . . 6  |-  ( om  ~<_  C  ->  ( ~P C  +c  ~P C ) 
~~  ~P C )
1816, 17syl 16 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  +c  ~P C
)  ~~  ~P C
)
19 simpr 461 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  B  ~<_  ~P ~P C )
20 gchdomtri 9003 . . . . 5  |-  ( ( ~P C  e. GCH  /\  ( ~P C  +c  ~P C )  ~~  ~P C  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C ) )
2112, 18, 19, 20syl3anc 1228 . . . 4  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C ) )
2221ex 434 . . 3  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C
) ) )
23 pwdom 7666 . . . . 5  |-  ( A  ~<_  C  ->  ~P A  ~<_  ~P C )
24 domtr 7565 . . . . . 6  |-  ( ( ~P A  ~<_  ~P C  /\  ~P C  ~<_  B )  ->  ~P A  ~<_  B )
2524ex 434 . . . . 5  |-  ( ~P A  ~<_  ~P C  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
262, 23, 253syl 20 . . . 4  |-  ( ph  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
271simprd 463 . . . 4  |-  ( ph  ->  ( B  ~<_  ~P C  ->  ~P A  ~<_  B ) )
2826, 27jaod 380 . . 3  |-  ( ph  ->  ( ( ~P C  ~<_  B  \/  B  ~<_  ~P C
)  ->  ~P A  ~<_  B ) )
2922, 28syld 44 . 2  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) )
3010, 29jca 532 1  |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447  (class class class)co 6282   omcom 6678    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512    +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-reu 2821  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: (None)
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