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Theorem isslw 15234
 Description: The property of being a Sylow subgroup. A Sylow -subgroup is a -group which has no proper supersets that are also -groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
isslw pSyl SubGrp SubGrp pGrp s
Distinct variable groups:   ,   ,   ,

Proof of Theorem isslw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-slw 15162 . . 3 pSyl SubGrp SubGrp pGrp s
21elmpt2cl 6280 . 2 pSyl
3 simp1 957 . . 3 SubGrp SubGrp pGrp s
4 subgrcl 14941 . . . 4 SubGrp
543ad2ant2 979 . . 3 SubGrp SubGrp pGrp s
63, 5jca 519 . 2 SubGrp SubGrp pGrp s
7 simpr 448 . . . . . . . . 9
87fveq2d 5724 . . . . . . . 8 SubGrp SubGrp
9 simpl 444 . . . . . . . . . . . 12
107oveq1d 6088 . . . . . . . . . . . 12 s s
119, 10breq12d 4217 . . . . . . . . . . 11 pGrp s pGrp s
1211anbi2d 685 . . . . . . . . . 10 pGrp s pGrp s
1312bibi1d 311 . . . . . . . . 9 pGrp s pGrp s
148, 13raleqbidv 2908 . . . . . . . 8 SubGrp pGrp s SubGrp pGrp s
158, 14rabeqbidv 2943 . . . . . . 7 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
16 fvex 5734 . . . . . . . 8 SubGrp
1716rabex 4346 . . . . . . 7 SubGrp SubGrp pGrp s
1815, 1, 17ovmpt2a 6196 . . . . . 6 pSyl SubGrp SubGrp pGrp s
1918eleq2d 2502 . . . . 5 pSyl SubGrp SubGrp pGrp s
20 sseq1 3361 . . . . . . . . 9
2120anbi1d 686 . . . . . . . 8 pGrp s pGrp s
22 eqeq1 2441 . . . . . . . 8
2321, 22bibi12d 313 . . . . . . 7 pGrp s pGrp s
2423ralbidv 2717 . . . . . 6 SubGrp pGrp s SubGrp pGrp s
2524elrab 3084 . . . . 5 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
2619, 25syl6bb 253 . . . 4 pSyl SubGrp SubGrp pGrp s
27 simpl 444 . . . . 5
2827biantrurd 495 . . . 4 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
2926, 28bitrd 245 . . 3 pSyl SubGrp SubGrp pGrp s
30 3anass 940 . . 3 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
3129, 30syl6bbr 255 . 2 pSyl SubGrp SubGrp pGrp s
322, 6, 31pm5.21nii 343 1 pSyl SubGrp SubGrp pGrp s
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  crab 2701   wss 3312   class class class wbr 4204  cfv 5446  (class class class)co 6073  cprime 13071   ↾s cress 13462  cgrp 14677  SubGrpcsubg 14930   pGrp cpgp 15157   pSyl cslw 15158 This theorem is referenced by:  slwprm  15235  slwsubg  15236  slwispgp  15237  pgpssslw  15240  subgslw  15242  fislw  15251 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-subg 14933  df-slw 15162
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