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Theorem slwpss 19656
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 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ¬ 𝑃 pGrp 𝑆)

Proof of Theorem slwpss
StepHypRef Expression
1 simp3 1138 . . 3 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
21pssned 4124 . 2 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
31pssssd 4123 . . . . 5 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
43biantrurd 532 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → (𝑃 pGrp 𝑆 ↔ (𝐻𝐾𝑃 pGrp 𝑆)))
5 slwispgp.1 . . . . . 6 𝑆 = (𝐺s 𝐾)
65slwispgp 19655 . . . . 5 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
763adant3 1132 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
84, 7bitrd 279 . . 3 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → (𝑃 pGrp 𝑆𝐻 = 𝐾))
98necon3bbid 2984 . 2 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → (¬ 𝑃 pGrp 𝑆𝐻𝐾))
102, 9mpbird 257 1 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ¬ 𝑃 pGrp 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wss 3976  wpss 3977   class class class wbr 5166  cfv 6575  (class class class)co 7450  s cress 17289  SubGrpcsubg 19162   pGrp cpgp 19570   pSyl cslw 19571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-subg 19165  df-slw 19575
This theorem is referenced by: (None)
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