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Metamath Proof Explorer

Theorem unieq

Description: Equality theorem for class union. Exercise 15 of TakeutiZaring p. 18. (Contributed by NM, 10-Aug-1993) (Proof shortened by Andrew Salmon, 29-Jun-2011) (Proof shortened by BJ, 13-Apr-2024)

		Ref	Expression
	Assertion	unieq	$⊢ A = B \to ⋃ A = ⋃ B$

Proof

Step	Hyp	Ref	Expression
1		eqimss	$⊢ A = B \to A \subseteq B$
2	1	unissd	$⊢ A = B \to ⋃ A \subseteq ⋃ B$
3		eqimss2	$⊢ A = B \to B \subseteq A$
4	3	unissd	$⊢ A = B \to ⋃ B \subseteq ⋃ A$
5	2 4	eqssd	$⊢ A = B \to ⋃ A = ⋃ B$