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Theorem simpg2nsg 19794
Description: A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Assertion
Ref Expression
simpg2nsg (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)

Proof of Theorem simpg2nsg
StepHypRef Expression
1 issimpg 19790 . 2 (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
21simprbi 497 1 (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105   class class class wbr 5092  cfv 6479  2oc2o 8361  cen 8801  Grpcgrp 18673  NrmSGrpcnsg 18846  SimpGrpcsimpg 19788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-simpg 19789
This theorem is referenced by:  trivnsimpgd  19795  simpgnsgd  19798
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