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Theorem gchaclem 8471
Description: Lemma for gchac 8474 (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 7044 . . . . . 6  |-  Rel  ~<_
43brrelex2i 4852 . . . . 5  |-  ( A  ~<_  C  ->  C  e.  _V )
52, 4syl 16 . . . 4  |-  ( ph  ->  C  e.  _V )
6 canth2g 7190 . . . 4  |-  ( C  e.  _V  ->  C  ~<  ~P C )
7 sdomdom 7064 . . . 4  |-  ( C 
~<  ~P C  ->  C  ~<_  ~P C )
85, 6, 73syl 19 . . 3  |-  ( ph  ->  C  ~<_  ~P C )
9 domtr 7089 . . 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 7089 . . . . . . . 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 8001 . . . . . 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 8430 . . . . 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 7188 . . . . 5  |-  ( A  ~<_  C  ->  ~P A  ~<_  ~P C )
24 domtr 7089 . . . . . 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 1717   _Vcvv 2892   ~Pcpw 3735   class class class wbr 4146   omcom 4778  (class class class)co 6013    ~~ cen 7035    ~<_ cdom 7036    ~< csdm 7037    +c ccda 7973  GCHcgch 8421
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-1o 6653  df-2o 6654  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-wdom 7453  df-card 7752  df-cda 7974  df-gch 8422
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