Theorem s1len

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

	Ref	Expression
Assertion	s1len	⊢ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1

Proof

Step	Hyp	Ref	Expression
1		df-s1	⊢ ⟨“ 𝐴 ”⟩ = { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ }
2	1	fveq2i	⊢ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = ( ♯ ‘ { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ } )
3		opex	⊢ ⟨ 0 , ( I ‘ 𝐴 ) ⟩ ∈ V
4		hashsng	⊢ ( ⟨ 0 , ( I ‘ 𝐴 ) ⟩ ∈ V → ( ♯ ‘ { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ } ) = 1 )
5	3 4	ax-mp	⊢ ( ♯ ‘ { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ } ) = 1
6	2 5	eqtri	⊢ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1