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Metamath Proof Explorer

Theorem tcid

Description: Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013)

		Ref	Expression
	Assertion	tcid	$⊢ A \in V \to A \subseteq TC (A)$

Proof

Step	Hyp	Ref	Expression
1		ssmin	$⊢ A \subseteq ⋂ \{x \| (A \subseteq x \land Tr x)\}$
2		tcvalg	$⊢ A \in V \to TC (A) = ⋂ \{x \| (A \subseteq x \land Tr x)\}$
3	1 2	sseqtrrid	$⊢ A \in V \to A \subseteq TC (A)$