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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 ∈ 𝑉 → 𝐴 ⊆ ( TC ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssmin	⊢ 𝐴 ⊆ ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) }
2		tcvalg	⊢ ( 𝐴 ∈ 𝑉 → ( TC ‘ 𝐴 ) = ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ) } )
3	1 2	sseqtrrid	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ⊆ ( TC ‘ 𝐴 ) )