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Theorem slwpss 19217
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 1137 . . 3 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
21pssned 4033 . 2 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
31pssssd 4032 . . . . 5 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → 𝐻𝐾)
43biantrurd 533 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → (𝑃 pGrp 𝑆 ↔ (𝐻𝐾𝑃 pGrp 𝑆)))
5 slwispgp.1 . . . . . 6 𝑆 = (𝐺s 𝐾)
65slwispgp 19216 . . . . 5 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
763adant3 1131 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
84, 7bitrd 278 . . 3 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → (𝑃 pGrp 𝑆𝐻 = 𝐾))
98necon3bbid 2981 . 2 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → (¬ 𝑃 pGrp 𝑆𝐻𝐾))
102, 9mpbird 256 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 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wss 3887  wpss 3888   class class class wbr 5074  cfv 6433  (class class class)co 7275  s cress 16941  SubGrpcsubg 18749   pGrp cpgp 19134   pSyl cslw 19135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-subg 18752  df-slw 19139
This theorem is referenced by: (None)
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