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Theorem slwpss 15016
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 957 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  C.  K
)
2 df-pss 3244 . . . 4  |-  ( H 
C.  K  <->  ( H  C_  K  /\  H  =/= 
K ) )
31, 2sylib 188 . . 3  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( H  C_  K  /\  H  =/=  K
) )
43simprd 449 . 2  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  =/=  K
)
51pssssd 3349 . . . . 5  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  H  C_  K
)
65biantrurd 494 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( P pGrp  S  <->  ( H  C_  K  /\  P pGrp  S ) ) )
7 slwispgp.1 . . . . . 6  |-  S  =  ( Gs  K )
87slwispgp 15015 . . . . 5  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )
)  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K
) )
983adant3 975 . . . 4  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( ( H 
C_  K  /\  P pGrp  S )  <->  H  =  K
) )
106, 9bitrd 244 . . 3  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( P pGrp  S  <->  H  =  K ) )
1110necon3bbid 2555 . 2  |-  ( ( H  e.  ( P pSyl 
G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  ( -.  P pGrp  S  <-> 
H  =/=  K ) )
124, 11mpbird 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521    C_ wss 3228    C. wpss 3229   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   ↾s cress 13240  SubGrpcsubg 14708   pGrp cpgp 14935   pSyl cslw 14936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-subg 14711  df-slw 14940
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