Theorem s1val

Description: Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015) (Revised by Mario Carneiro, 26-Feb-2016)

	Ref	Expression
Assertion	s1val	⊢ ( 𝐴 ∈ 𝑉 → ⟨“ 𝐴 ”⟩ = { ⟨ 0 , 𝐴 ⟩ } )

Proof

Step	Hyp	Ref	Expression
1		df-s1	⊢ ⟨“ 𝐴 ”⟩ = { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ }
2		fvi	⊢ ( 𝐴 ∈ 𝑉 → ( I ‘ 𝐴 ) = 𝐴 )
3	2	opeq2d	⊢ ( 𝐴 ∈ 𝑉 → ⟨ 0 , ( I ‘ 𝐴 ) ⟩ = ⟨ 0 , 𝐴 ⟩ )
4	3	sneqd	⊢ ( 𝐴 ∈ 𝑉 → { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ } = { ⟨ 0 , 𝐴 ⟩ } )
5	1 4	syl5eq	⊢ ( 𝐴 ∈ 𝑉 → ⟨“ 𝐴 ”⟩ = { ⟨ 0 , 𝐴 ⟩ } )