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Metamath Proof Explorer

Theorem unictb

Description: The countable union of countable sets is countable. Theorem 6Q of Enderton p. 159. See iunctb for indexed union version. (Contributed by NM, 26-Mar-2006)

		Ref	Expression
	Assertion	unictb	$⊢ (A ≼ ω \land \forall x \in A x ≼ ω) \to ⋃ A ≼ ω$

Proof

Step	Hyp	Ref	Expression
1		uniiun	$⊢ ⋃ A = ⋃_{x \in A} x$
2		iunctb	$⊢ (A ≼ ω \land \forall x \in A x ≼ ω) \to ⋃_{x \in A} x ≼ ω$
3	1 2	eqbrtrid	$⊢ (A ≼ ω \land \forall x \in A x ≼ ω) \to ⋃ A ≼ ω$