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Metamath Proof Explorer

Theorem trivsubgsnd

Description: The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023)

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

Proof

Step Hyp Ref Expression
1 trivsubgsnd.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 trivsubgsnd.2 ⊢ 0 = ( 0g ‘ 𝐺 )
3 trivsubgsnd.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
4 trivsubgsnd.4 ⊢ ( 𝜑 → 𝐵 = { 0 } )
5 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝐺 ∈ Grp )
6 4 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝐵 = { 0 } )
7 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) )
8 1 2 5 6 7 trivsubgd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑥 = 𝐵 )
9 velsn ⊢ ( 𝑥 ∈ { 𝐵 } ↔ 𝑥 = 𝐵 )
10 8 9 sylibr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑥 ∈ { 𝐵 } )
11 10 ex ⊢ ( 𝜑 → ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) → 𝑥 ∈ { 𝐵 } ) )
12 11 ssrdv ⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) ⊆ { 𝐵 } )
13 1 subgid ⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) )
14 3 13 syl ⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) )
15 14 snssd ⊢ ( 𝜑 → { 𝐵 } ⊆ ( SubGrp ‘ 𝐺 ) )
16 12 15 eqssd ⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) = { 𝐵 } )