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	$$⊢A\in V\to A\subseteq \mathrm{TC}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssmin	$$⊢A\subseteq \bigcap \left\{x|\left(A\subseteq x\wedge \mathrm{Tr}x\right)\right\}$$
2		tcvalg	$$⊢A\in V\to \mathrm{TC}\left(A\right)=\bigcap \left\{x|\left(A\subseteq x\wedge \mathrm{Tr}x\right)\right\}$$
3	1 2	sseqtrrid	$$⊢A\in V\to A\subseteq \mathrm{TC}\left(A\right)$$