Theorem s5cld

Description: A length 5 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	$$⊢\phi \to C\in X$$
	s4cld.4	$$⊢\phi \to D\in X$$
	s5cld.5	$$⊢\phi \to E\in X$$
Assertion	s5cld	$$⊢\phi \to ⟨“ABCDE”⟩\in \mathrm{Word}X$$

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		s4cld.4	$$⊢\phi \to D\in X$$
5		s5cld.5	$$⊢\phi \to E\in X$$
6		df-s5	$$⊢⟨“ABCDE”⟩=⟨“ABCD”⟩\mathrm{++}⟨“E”⟩$$
7	1 2 3 4	s4cld	$$⊢\phi \to ⟨“ABCD”⟩\in \mathrm{Word}X$$
8	6 7 5	cats1cld	$$⊢\phi \to ⟨“ABCDE”⟩\in \mathrm{Word}X$$