This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ixpssixp

Description: Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	ixpssixp.1	$⊢ Ⅎ x φ$
	ixpssixp.2	$⊢ (φ \land x \in A) \to B \subseteq C$
Assertion	ixpssixp	$⊢ φ \to \underset{x \in A}{⨉} B \subseteq \underset{x \in A}{⨉} C$

Step	Hyp	Ref	Expression
1		ixpssixp.1	$⊢ Ⅎ x φ$
2		ixpssixp.2	$⊢ (φ \land x \in A) \to B \subseteq C$
3	2	ex	$⊢ φ \to (x \in A \to B \subseteq C)$
4	1 3	ralrimi	$⊢ φ \to \forall x \in A B \subseteq C$
5		ss2ixp	$⊢ \forall x \in A B \subseteq C \to \underset{x \in A}{⨉} B \subseteq \underset{x \in A}{⨉} C$
6	4 5	syl	$⊢ φ \to \underset{x \in A}{⨉} B \subseteq \underset{x \in A}{⨉} C$