Theorem tpssd

Description: Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025)

	Ref	Expression
Hypotheses	tpssd.1	$$⊢\phi \to A\in D$$
	tpssd.2	$$⊢\phi \to B\in D$$
	tpssd.3	$$⊢\phi \to C\in D$$
Assertion	tpssd	$$⊢\phi \to \left\{A,B,C\right\}\subseteq D$$

Proof

Step	Hyp	Ref	Expression
1		tpssd.1	$$⊢\phi \to A\in D$$
2		tpssd.2	$$⊢\phi \to B\in D$$
3		tpssd.3	$$⊢\phi \to C\in D$$
4		tpssi	$$⊢\left(A\in D\wedge B\in D\wedge C\in D\right)\to \left\{A,B,C\right\}\subseteq D$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left\{A,B,C\right\}\subseteq D$$