Theorem simpg2nsg 20064

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

Step	Hyp	Ref	Expression
1		issimpg 20060	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simprbi 498	1 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)

Syntax hints:   → wi 4   ∈ wcel 2119   class class class wbr 5072  ‘cfv 6485  2_oc2o 8389   ≈ cen 8880  Grpcgrp 18900  NrmSGrpcnsg 19088  SimpGrpcsimpg 20058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-simpg 20059

This theorem is referenced by:  trivnsimpgd  20065  simpgnsgd  20068