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Theorem gchaclem 8292
Description: Lemma for gchac 8295 (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 445 . . 3  |-  ( ph  ->  A  ~<_  C )
3 reldom 6869 . . . . . 6  |-  Rel  ~<_
43brrelex2i 4730 . . . . 5  |-  ( A  ~<_  C  ->  C  e.  _V )
52, 4syl 15 . . . 4  |-  ( ph  ->  C  e.  _V )
6 canth2g 7015 . . . 4  |-  ( C  e.  _V  ->  C  ~<  ~P C )
7 sdomdom 6889 . . . 4  |-  ( C 
~<  ~P C  ->  C  ~<_  ~P C )
85, 6, 73syl 18 . . 3  |-  ( ph  ->  C  ~<_  ~P C )
9 domtr 6914 . . 3  |-  ( ( A  ~<_  C  /\  C  ~<_  ~P C )  ->  A  ~<_  ~P C )
102, 8, 9syl2anc 642 . 2  |-  ( ph  ->  A  ~<_  ~P C )
11 gchaclem.3 . . . . . 6  |-  ( ph  ->  ~P C  e. GCH )
1211adantr 451 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ~P C  e. GCH )
13 gchaclem.1 . . . . . . . 8  |-  ( ph  ->  om  ~<_  A )
14 domtr 6914 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  C )  ->  om  ~<_  C )
1513, 2, 14syl2anc 642 . . . . . . 7  |-  ( ph  ->  om  ~<_  C )
1615adantr 451 . . . . . 6  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  om  ~<_  C )
17 pwcdaidm 7821 . . . . . 6  |-  ( om  ~<_  C  ->  ( ~P C  +c  ~P C ) 
~~  ~P C )
1816, 17syl 15 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  +c  ~P C
)  ~~  ~P C
)
19 simpr 447 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  B  ~<_  ~P ~P C )
20 gchdomtri 8251 . . . . 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 1182 . . . 4  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C ) )
2221ex 423 . . 3  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C
) ) )
23 pwdom 7013 . . . . 5  |-  ( A  ~<_  C  ->  ~P A  ~<_  ~P C )
24 domtr 6914 . . . . . 6  |-  ( ( ~P A  ~<_  ~P C  /\  ~P C  ~<_  B )  ->  ~P A  ~<_  B )
2524ex 423 . . . . 5  |-  ( ~P A  ~<_  ~P C  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
262, 23, 253syl 18 . . . 4  |-  ( ph  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
271simprd 449 . . . 4  |-  ( ph  ->  ( B  ~<_  ~P C  ->  ~P A  ~<_  B ) )
2826, 27jaod 369 . . 3  |-  ( ph  ->  ( ( ~P C  ~<_  B  \/  B  ~<_  ~P C
)  ->  ~P A  ~<_  B ) )
2922, 28syld 40 . 2  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) )
3010, 29jca 518 1  |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023   omcom 4656  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862    +c ccda 7793  GCHcgch 8242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-wdom 7273  df-card 7572  df-cda 7794  df-gch 8243
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