Theorem s3cld

Description: A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016)

	Ref	Expression
Hypotheses	s2cld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	s2cld.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
	s3cld.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
Assertion	s3cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		s2cld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
2		s2cld.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
3		s3cld.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
4		df-s3	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ = ( ⟨“ 𝐴 𝐵 ”⟩ ++ ⟨“ 𝐶 ”⟩ )
5	1 2	s2cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 ”⟩ ∈ Word 𝑋 )
6	4 5 3	cats1cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word 𝑋 )