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Theorem isslw 16754
 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 16682 . . 3 pSyl SubGrp SubGrp pGrp s
21elmpt2cl 6516 . 2 pSyl
3 simp1 996 . . 3 SubGrp SubGrp pGrp s
4 subgrcl 16332 . . . 4 SubGrp
543ad2ant2 1018 . . 3 SubGrp SubGrp pGrp s
63, 5jca 532 . 2 SubGrp SubGrp pGrp s
7 simpr 461 . . . . . . . . 9
87fveq2d 5876 . . . . . . . 8 SubGrp SubGrp
9 simpl 457 . . . . . . . . . . . 12
107oveq1d 6311 . . . . . . . . . . . 12 s s
119, 10breq12d 4469 . . . . . . . . . . 11 pGrp s pGrp s
1211anbi2d 703 . . . . . . . . . 10 pGrp s pGrp s
1312bibi1d 319 . . . . . . . . 9 pGrp s pGrp s
148, 13raleqbidv 3068 . . . . . . . 8 SubGrp pGrp s SubGrp pGrp s
158, 14rabeqbidv 3104 . . . . . . 7 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
16 fvex 5882 . . . . . . . 8 SubGrp
1716rabex 4607 . . . . . . 7 SubGrp SubGrp pGrp s
1815, 1, 17ovmpt2a 6432 . . . . . 6 pSyl SubGrp SubGrp pGrp s
1918eleq2d 2527 . . . . 5 pSyl SubGrp SubGrp pGrp s
20 sseq1 3520 . . . . . . . . 9
2120anbi1d 704 . . . . . . . 8 pGrp s pGrp s
22 eqeq1 2461 . . . . . . . 8
2321, 22bibi12d 321 . . . . . . 7 pGrp s pGrp s
2423ralbidv 2896 . . . . . 6 SubGrp pGrp s SubGrp pGrp s
2524elrab 3257 . . . . 5 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
2619, 25syl6bb 261 . . . 4 pSyl SubGrp SubGrp pGrp s
27 simpl 457 . . . . 5
2827biantrurd 508 . . . 4 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
2926, 28bitrd 253 . . 3 pSyl SubGrp SubGrp pGrp s
30 3anass 977 . . 3 SubGrp SubGrp pGrp s SubGrp SubGrp pGrp s
3129, 30syl6bbr 263 . 2 pSyl SubGrp SubGrp pGrp s
322, 6, 31pm5.21nii 353 1 pSyl SubGrp SubGrp pGrp s
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   w3a 973   wceq 1395   wcel 1819  wral 2807  crab 2811   wss 3471   class class class wbr 4456  cfv 5594  (class class class)co 6296  cprime 14228   ↾s cress 14644  cgrp 16179  SubGrpcsubg 16321   pGrp cpgp 16677   pSyl cslw 16678 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-subg 16324  df-slw 16682 This theorem is referenced by:  slwprm  16755  slwsubg  16756  slwispgp  16757  pgpssslw  16760  subgslw  16762  fislw  16771
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