Theorem s2cld

Description: A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016)

	Ref	Expression
Hypotheses	s2cld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	s2cld.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
Assertion	s2cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 ”⟩ ∈ Word 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		s2cld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
2		s2cld.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
3		df-s2	⊢ ⟨“ 𝐴 𝐵 ”⟩ = ( ⟨“ 𝐴 ”⟩ ++ ⟨“ 𝐵 ”⟩ )
4	1	s1cld	⊢ ( 𝜑 → ⟨“ 𝐴 ”⟩ ∈ Word 𝑋 )
5	3 4 2	cats1cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 ”⟩ ∈ Word 𝑋 )