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Theorem gchaclem 9045
Description: Lemma for gchac 9048 (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 457 . . 3  |-  ( ph  ->  A  ~<_  C )
3 reldom 7515 . . . . . 6  |-  Rel  ~<_
43brrelex2i 5030 . . . . 5  |-  ( A  ~<_  C  ->  C  e.  _V )
52, 4syl 16 . . . 4  |-  ( ph  ->  C  e.  _V )
6 canth2g 7664 . . . 4  |-  ( C  e.  _V  ->  C  ~<  ~P C )
7 sdomdom 7536 . . . 4  |-  ( C 
~<  ~P C  ->  C  ~<_  ~P C )
85, 6, 73syl 20 . . 3  |-  ( ph  ->  C  ~<_  ~P C )
9 domtr 7561 . . 3  |-  ( ( A  ~<_  C  /\  C  ~<_  ~P C )  ->  A  ~<_  ~P C )
102, 8, 9syl2anc 659 . 2  |-  ( ph  ->  A  ~<_  ~P C )
11 gchaclem.3 . . . . . 6  |-  ( ph  ->  ~P C  e. GCH )
1211adantr 463 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ~P C  e. GCH )
13 gchaclem.1 . . . . . . . 8  |-  ( ph  ->  om  ~<_  A )
14 domtr 7561 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  C )  ->  om  ~<_  C )
1513, 2, 14syl2anc 659 . . . . . . 7  |-  ( ph  ->  om  ~<_  C )
1615adantr 463 . . . . . 6  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  om  ~<_  C )
17 pwcdaidm 8566 . . . . . 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 459 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  B  ~<_  ~P ~P C )
20 gchdomtri 8996 . . . . 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 1226 . . . 4  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C ) )
2221ex 432 . . 3  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C
) ) )
23 pwdom 7662 . . . . 5  |-  ( A  ~<_  C  ->  ~P A  ~<_  ~P C )
24 domtr 7561 . . . . . 6  |-  ( ( ~P A  ~<_  ~P C  /\  ~P C  ~<_  B )  ->  ~P A  ~<_  B )
2524ex 432 . . . . 5  |-  ( ~P A  ~<_  ~P C  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
262, 23, 253syl 20 . . . 4  |-  ( ph  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
271simprd 461 . . . 4  |-  ( ph  ->  ( B  ~<_  ~P C  ->  ~P A  ~<_  B ) )
2826, 27jaod 378 . . 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 530 1  |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    e. wcel 1823   _Vcvv 3106   ~Pcpw 3999   class class class wbr 4439  (class class class)co 6270   omcom 6673    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508    +c ccda 8538  GCHcgch 8987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-1o 7122  df-2o 7123  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-wdom 7977  df-card 8311  df-cda 8539  df-gch 8988
This theorem is referenced by: (None)
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