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Theorem issubgo 21896
 Description: The predicate "is a subgroup of ." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
issubgo

Proof of Theorem issubgo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 inss2 3564 . . . . . . 7
2 pwexg 4386 . . . . . . 7
3 ssexg 4352 . . . . . . 7
41, 2, 3sylancr 646 . . . . . 6
5 pweq 3804 . . . . . . . 8
65ineq2d 3544 . . . . . . 7
7 df-subgo 21895 . . . . . . 7
86, 7fvmptg 5807 . . . . . 6
94, 8mpdan 651 . . . . 5
109eleq2d 2505 . . . 4
11 elin 3532 . . . . 5
12 elpwg 3808 . . . . . 6
1312pm5.32i 620 . . . . 5
1411, 13bitri 242 . . . 4
1510, 14syl6bb 254 . . 3
1615pm5.32i 620 . 2
177dmmptss 5369 . . . 4
18 elfvdm 5760 . . . 4
1917, 18sseldi 3348 . . 3
2019pm4.71ri 616 . 2
21 3anass 941 . 2
2216, 20, 213bitr4i 270 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  cvv 2958   cin 3321   wss 3322  cpw 3801   cdm 4881  cfv 5457  cgr 21779  csubgo 21894 This theorem is referenced by:  subgores  21897  subgoid  21900  subgoinv  21901  issubgoi  21903  subgoablo  21904  ghsubgolem  21963  hhssabloi  22767  ghomfo  25107  ghomgsg  25109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-subgo 21895
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