Theorem 1nsgtrivd

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

	Ref	Expression
Hypotheses	1nsgtrivd.1	$$⊢B={\mathrm{Base}}_G$$
	1nsgtrivd.2	$$⊢\underset{\dot{}}{0}=0_G$$
	1nsgtrivd.3	$$⊢\phi \to G\in \mathrm{Grp}$$
	1nsgtrivd.4	$$⊢\phi \to \mathrm{NrmSGrp}\left(G\right)\approx 1_{𝑜}$$
Assertion	1nsgtrivd	$$⊢\phi \to B=\left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		1nsgtrivd.1	$$⊢B={\mathrm{Base}}_G$$
2		1nsgtrivd.2	$$⊢\underset{\dot{}}{0}=0_G$$
3		1nsgtrivd.3	$$⊢\phi \to G\in \mathrm{Grp}$$
4		1nsgtrivd.4	$$⊢\phi \to \mathrm{NrmSGrp}\left(G\right)\approx 1_{𝑜}$$
5	1	nsgid	$$⊢G\in \mathrm{Grp}\to B\in \mathrm{NrmSGrp}\left(G\right)$$
6	3 5	syl	$$⊢\phi \to B\in \mathrm{NrmSGrp}\left(G\right)$$
7	2	0nsg	$$⊢G\in \mathrm{Grp}\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{NrmSGrp}\left(G\right)$$
8	3 7	syl	$$⊢\phi \to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{NrmSGrp}\left(G\right)$$
9		en1eqsn	$$⊢\left(\left\{\underset{\dot{}}{0}\right\}\in \mathrm{NrmSGrp}\left(G\right)\wedge \mathrm{NrmSGrp}\left(G\right)\approx 1_{𝑜}\right)\to \mathrm{NrmSGrp}\left(G\right)=\left\{\left\{\underset{\dot{}}{0}\right\}\right\}$$
10	8 4 9	syl2anc	$$⊢\phi \to \mathrm{NrmSGrp}\left(G\right)=\left\{\left\{\underset{\dot{}}{0}\right\}\right\}$$
11	6 10	eleqtrd	$$⊢\phi \to B\in \left\{\left\{\underset{\dot{}}{0}\right\}\right\}$$
12		snex	$$⊢\left\{\underset{\dot{}}{0}\right\}\in \mathrm{V}$$
13		elsn2g	$$⊢\left\{\underset{\dot{}}{0}\right\}\in \mathrm{V}\to \left(B\in \left\{\left\{\underset{\dot{}}{0}\right\}\right\}\leftrightarrow B=\left\{\underset{\dot{}}{0}\right\}\right)$$
14	12 13	mp1i	$$⊢\phi \to \left(B\in \left\{\left\{\underset{\dot{}}{0}\right\}\right\}\leftrightarrow B=\left\{\underset{\dot{}}{0}\right\}\right)$$
15	11 14	mpbid	$$⊢\phi \to B=\left\{\underset{\dot{}}{0}\right\}$$