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Theorem gchaclem 8537
Description: Lemma for gchac 8540 (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 446 . . 3  |-  ( ph  ->  A  ~<_  C )
3 reldom 7107 . . . . . 6  |-  Rel  ~<_
43brrelex2i 4911 . . . . 5  |-  ( A  ~<_  C  ->  C  e.  _V )
52, 4syl 16 . . . 4  |-  ( ph  ->  C  e.  _V )
6 canth2g 7253 . . . 4  |-  ( C  e.  _V  ->  C  ~<  ~P C )
7 sdomdom 7127 . . . 4  |-  ( C 
~<  ~P C  ->  C  ~<_  ~P C )
85, 6, 73syl 19 . . 3  |-  ( ph  ->  C  ~<_  ~P C )
9 domtr 7152 . . 3  |-  ( ( A  ~<_  C  /\  C  ~<_  ~P C )  ->  A  ~<_  ~P C )
102, 8, 9syl2anc 643 . 2  |-  ( ph  ->  A  ~<_  ~P C )
11 gchaclem.3 . . . . . 6  |-  ( ph  ->  ~P C  e. GCH )
1211adantr 452 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ~P C  e. GCH )
13 gchaclem.1 . . . . . . . 8  |-  ( ph  ->  om  ~<_  A )
14 domtr 7152 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  C )  ->  om  ~<_  C )
1513, 2, 14syl2anc 643 . . . . . . 7  |-  ( ph  ->  om  ~<_  C )
1615adantr 452 . . . . . 6  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  om  ~<_  C )
17 pwcdaidm 8067 . . . . . 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 448 . . . . 5  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  B  ~<_  ~P ~P C )
20 gchdomtri 8496 . . . . 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 1184 . . . 4  |-  ( (
ph  /\  B  ~<_  ~P ~P C )  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C ) )
2221ex 424 . . 3  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ( ~P C  ~<_  B  \/  B  ~<_  ~P C
) ) )
23 pwdom 7251 . . . . 5  |-  ( A  ~<_  C  ->  ~P A  ~<_  ~P C )
24 domtr 7152 . . . . . 6  |-  ( ( ~P A  ~<_  ~P C  /\  ~P C  ~<_  B )  ->  ~P A  ~<_  B )
2524ex 424 . . . . 5  |-  ( ~P A  ~<_  ~P C  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
262, 23, 253syl 19 . . . 4  |-  ( ph  ->  ( ~P C  ~<_  B  ->  ~P A  ~<_  B ) )
271simprd 450 . . . 4  |-  ( ph  ->  ( B  ~<_  ~P C  ->  ~P A  ~<_  B ) )
2826, 27jaod 370 . . 3  |-  ( ph  ->  ( ( ~P C  ~<_  B  \/  B  ~<_  ~P C
)  ->  ~P A  ~<_  B ) )
2922, 28syld 42 . 2  |-  ( ph  ->  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) )
3010, 29jca 519 1  |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    e. wcel 1725   _Vcvv 2948   ~Pcpw 3791   class class class wbr 4204   omcom 4837  (class class class)co 6073    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100    +c ccda 8039  GCHcgch 8487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-2o 6717  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-wdom 7519  df-card 7818  df-cda 8040  df-gch 8488
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