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Theorem slwpss 19560
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 1136 . . 3 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
21pssned 4094 . 2 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
31pssssd 4093 . . . . 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 19559 . . . . 5 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
763adant3 1130 . . . 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 2973 . 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wss 3944  wpss 3945   class class class wbr 5142  cfv 6542  (class class class)co 7414  s cress 17202  SubGrpcsubg 19068   pGrp cpgp 19474   pSyl cslw 19475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-subg 19071  df-slw 19479
This theorem is referenced by: (None)
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