Theorem s3cld

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

	Ref	Expression
Hypotheses	s2cld.1	$$⊢\phi \to A\in X$$
	s2cld.2	$$⊢\phi \to B\in X$$
	s3cld.3	$$⊢\phi \to C\in X$$
Assertion	s3cld	$$⊢\phi \to ⟨“ABC”⟩\in \mathrm{Word}X$$

Proof

Step	Hyp	Ref	Expression
1		s2cld.1	$$⊢\phi \to A\in X$$
2		s2cld.2	$$⊢\phi \to B\in X$$
3		s3cld.3	$$⊢\phi \to C\in X$$
4		df-s3	$$⊢⟨“ABC”⟩=⟨“AB”⟩\mathrm{++}⟨“C”⟩$$
5	1 2	s2cld	$$⊢\phi \to ⟨“AB”⟩\in \mathrm{Word}X$$
6	4 5 3	cats1cld	$$⊢\phi \to ⟨“ABC”⟩\in \mathrm{Word}X$$