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Metamath Proof Explorer

Theorem iunxunsn

Description: Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023)

		Ref	Expression
	Hypothesis	iunxunsn.1	$⊢ x = X \to B = C$
	Assertion	iunxunsn	$⊢ X \in V \to ⋃_{x \in A \cup \{X\}} B = ⋃_{x \in A} B \cup C$

Step	Hyp	Ref	Expression
1		iunxunsn.1	$⊢ x = X \to B = C$
2		iunxun	$⊢ ⋃_{x \in A \cup \{X\}} B = ⋃_{x \in A} B \cup ⋃_{x \in \{X\}} B$
3	1	iunxsng	$⊢ X \in V \to ⋃_{x \in \{X\}} B = C$
4	3	uneq2d	$⊢ X \in V \to ⋃_{x \in A} B \cup ⋃_{x \in \{X\}} B = ⋃_{x \in A} B \cup C$
5	2 4	eqtrid	$⊢ X \in V \to ⋃_{x \in A \cup \{X\}} B = ⋃_{x \in A} B \cup C$