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Theorem slwpss 19526

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 = (𝐺 ↾_s 𝐾)

**Assertion**
Ref	Expression
slwpss	⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆)

**Proof of Theorem slwpss**
Step	Hyp	Ref	Expression
1		simp3 1138	. . 3 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → 𝐻 ⊊ 𝐾)
2	1	pssned 4060	. 2 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → 𝐻 ≠ 𝐾)
3	1	pssssd 4059	. . . . 5 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → 𝐻 ⊆ 𝐾)
4	3	biantrurd 532	. . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → (𝑃 pGrp 𝑆 ↔ (𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆)))
5		slwispgp.1	. . . . . 6 ⊢ 𝑆 = (𝐺 ↾_s 𝐾)
6	5	slwispgp 19525	. . . . 5 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
7	6	3adant3 1132	. . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
8	4, 7	bitrd 279	. . 3 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → (𝑃 pGrp 𝑆 ↔ 𝐻 = 𝐾))
9	8	necon3bbid 2962	. 2 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → (¬ 𝑃 pGrp 𝑆 ↔ 𝐻 ≠ 𝐾))
10	2, 9	mpbird 257	1 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3911 ⊊ wpss 3912 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ↾_s cress 17176 SubGrpcsubg 19034 pGrp cpgp 19440 pSyl cslw 19441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-subg 19037 df-slw 19445

This theorem is referenced by: (None)

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