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Theorem slwpss 15201
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 959 . . 3  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  C.  K
)
21pssned 3405 . 2  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  =/=  K
)
31pssssd 3404 . . . . 5  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  C_  K
)
43biantrurd 495 . . . 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 15200 . . . . 5  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
763adant3 977 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( ( H 
C_  K  /\  P pGrp  S )  <->  H  =  K
) )
84, 7bitrd 245 . . 3  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( P pGrp  S  <->  H  =  K ) )
98necon3bbid 2601 . 2  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( -.  P pGrp  S  <-> 
H  =/=  K ) )
102, 9mpbird 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280    C. wpss 3281   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   ↾s cress 13425  SubGrpcsubg 14893   pGrp cpgp 15120   pSyl cslw 15121
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-subg 14896  df-slw 15125
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