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Theorem slwpss 15251
Description: A proper superset of a Sylow subgroup is not a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
slwispgp.1  |-  S  =  ( Gs  K )
Assertion
Ref Expression
slwpss  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  -.  P pGrp  S )

Proof of Theorem slwpss
StepHypRef Expression
1 simp3 960 . . 3  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  C.  K
)
21pssned 3447 . 2  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  =/=  K
)
31pssssd 3446 . . . . 5  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  C_  K
)
43biantrurd 496 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( P pGrp  S  <->  ( H  C_  K  /\  P pGrp  S ) ) )
5 slwispgp.1 . . . . . 6  |-  S  =  ( Gs  K )
65slwispgp 15250 . . . . 5  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
763adant3 978 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( ( H 
C_  K  /\  P pGrp  S )  <->  H  =  K
) )
84, 7bitrd 246 . . 3  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( P pGrp  S  <->  H  =  K ) )
98necon3bbid 2637 . 2  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( -.  P pGrp  S  <-> 
H  =/=  K ) )
102, 9mpbird 225 1  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  -.  P pGrp  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322    C. wpss 3323   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   ↾s cress 13475  SubGrpcsubg 14943   pGrp cpgp 15170   pSyl cslw 15171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-subg 14946  df-slw 15175
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