Theorem rabsstp

Description: Conditions for a restricted class abstraction to be a subset of an unordered triple. (Contributed by Thierry Arnoux, 6-Jul-2025)

	Ref	Expression
Assertion	rabsstp	$$⊢\left\{x\in V|\phi \right\}\subseteq \left\{X,Y,Z\right\}\leftrightarrow \forall x\in V\left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-rab	$$⊢\left\{x\in V|\phi \right\}=\left\{x|\left(x\in V\wedge \phi \right)\right\}$$
2		dftp2	$$⊢\left\{X,Y,Z\right\}=\left\{x|\left(x=X\vee x=Y\vee x=Z\right)\right\}$$
3	1 2	sseq12i	$$⊢\left\{x\in V|\phi \right\}\subseteq \left\{X,Y,Z\right\}\leftrightarrow \left\{x|\left(x\in V\wedge \phi \right)\right\}\subseteq \left\{x|\left(x=X\vee x=Y\vee x=Z\right)\right\}$$
4		ss2ab	$$⊢\left\{x|\left(x\in V\wedge \phi \right)\right\}\subseteq \left\{x|\left(x=X\vee x=Y\vee x=Z\right)\right\}\leftrightarrow \forall x\left(\left(x\in V\wedge \phi \right)\to \left(x=X\vee x=Y\vee x=Z\right)\right)$$
5		impexp	$$⊢\left(\left(x\in V\wedge \phi \right)\to \left(x=X\vee x=Y\vee x=Z\right)\right)\leftrightarrow \left(x\in V\to \left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)\right)$$
6	5	albii	$$⊢\forall x\left(\left(x\in V\wedge \phi \right)\to \left(x=X\vee x=Y\vee x=Z\right)\right)\leftrightarrow \forall x\left(x\in V\to \left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)\right)$$
7		df-ral	$$⊢\forall x\in V\left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)\leftrightarrow \forall x\left(x\in V\to \left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)\right)$$
8	6 7	bitr4i	$$⊢\forall x\left(\left(x\in V\wedge \phi \right)\to \left(x=X\vee x=Y\vee x=Z\right)\right)\leftrightarrow \forall x\in V\left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)$$
9	3 4 8	3bitri	$$⊢\left\{x\in V|\phi \right\}\subseteq \left\{X,Y,Z\right\}\leftrightarrow \forall x\in V\left(\phi \to \left(x=X\vee x=Y\vee x=Z\right)\right)$$