Theorem simpg2nsg 19196

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 19192	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simprbi 499	1 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5047  ‘cfv 6336  2_oc2o 8077   ≈ cen 8487  Grpcgrp 18081  NrmSGrpcnsg 18252  SimpGrpcsimpg 19190

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3142  df-v 3483  df-dif 3922  df-un 3924  df-in 3926  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-iota 6295  df-fv 6344  df-simpg 19191

This theorem is referenced by:  trivnsimpgd  19197  simpgnsgd  19200