Theorem 1nsgtrivd

Description: A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	1nsgtrivd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	1nsgtrivd.2	⊢ 0 = ( 0g ‘ 𝐺 )
	1nsgtrivd.3	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	1nsgtrivd.4	⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ≈ 1o )
Assertion	1nsgtrivd	⊢ ( 𝜑 → 𝐵 = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		1nsgtrivd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		1nsgtrivd.2	⊢ 0 = ( 0g ‘ 𝐺 )
3		1nsgtrivd.3	⊢ ( 𝜑 → 𝐺 ∈ Grp )
4		1nsgtrivd.4	⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ≈ 1o )
5	1	nsgid	⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) )
6	3 5	syl	⊢ ( 𝜑 → 𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) )
7	2	0nsg	⊢ ( 𝐺 ∈ Grp → { 0 } ∈ ( NrmSGrp ‘ 𝐺 ) )
8	3 7	syl	⊢ ( 𝜑 → { 0 } ∈ ( NrmSGrp ‘ 𝐺 ) )
9		en1eqsn	⊢ ( ( { 0 } ∈ ( NrmSGrp ‘ 𝐺 ) ∧ ( NrmSGrp ‘ 𝐺 ) ≈ 1o ) → ( NrmSGrp ‘ 𝐺 ) = { { 0 } } )
10	8 4 9	syl2anc	⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) = { { 0 } } )
11	6 10	eleqtrd	⊢ ( 𝜑 → 𝐵 ∈ { { 0 } } )
12		snex	⊢ { 0 } ∈ V
13		elsn2g	⊢ ( { 0 } ∈ V → ( 𝐵 ∈ { { 0 } } ↔ 𝐵 = { 0 } ) )
14	12 13	mp1i	⊢ ( 𝜑 → ( 𝐵 ∈ { { 0 } } ↔ 𝐵 = { 0 } ) )
15	11 14	mpbid	⊢ ( 𝜑 → 𝐵 = { 0 } )