Theorem intmin2

Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003)

	Ref	Expression
Hypothesis	intmin2.1	⊢ 𝐴 ∈ V
Assertion	intmin2	⊢ ∩ { 𝑥 ∣ 𝐴 ⊆ 𝑥 } = 𝐴

Proof

Step	Hyp	Ref	Expression
1		intmin2.1	⊢ 𝐴 ∈ V
2		rabab	⊢ { 𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥 } = { 𝑥 ∣ 𝐴 ⊆ 𝑥 }
3	2	inteqi	⊢ ∩ { 𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥 } = ∩ { 𝑥 ∣ 𝐴 ⊆ 𝑥 }
4		intmin	⊢ ( 𝐴 ∈ V → ∩ { 𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥 } = 𝐴 )
5	1 4	ax-mp	⊢ ∩ { 𝑥 ∈ V ∣ 𝐴 ⊆ 𝑥 } = 𝐴
6	3 5	eqtr3i	⊢ ∩ { 𝑥 ∣ 𝐴 ⊆ 𝑥 } = 𝐴