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 497	1 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492  2_oc2o 8392   ≈ cen 8883  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-simpg 20059

This theorem is referenced by:  trivnsimpgd  20065  simpgnsgd  20068