Metamath Proof Explorer

Theorem slwpss

Description: A proper superset of a Sylow subgroup is not a P -group. (Contributed by Mario Carneiro, 16-Jan-2015)

Ref Expression
Hypothesis slwispgp.1 
|- S = ( G |`s K )
Assertion slwpss 
|- ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> -. P pGrp S )

Proof

Step Hyp Ref Expression
1 slwispgp.1 
 |-  S = ( G |`s K )
2 simp3 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> H C. K )
3 2 pssned 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> H =/= K )
4 2 pssssd 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> H C_ K )
5 4 biantrurd 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> ( P pGrp S <-> ( H C_ K /\ P pGrp S ) ) )
6 1 slwispgp 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )
7 6 3adant3 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )
8 5 7 bitrd 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> ( P pGrp S <-> H = K ) )
9 8 necon3bbid 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> ( -. P pGrp S <-> H =/= K ) )
10 3 9 mpbird 
 |-  ( ( H e. ( P pSyl G ) /\ K e. ( SubGrp ` G ) /\ H C. K ) -> -. P pGrp S )