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Metamath Proof Explorer

Theorem int0

Description: The intersection of the empty set is the universal class. Exercise 2 of TakeutiZaring p. 44. (Contributed by NM, 18-Aug-1993) (Proof shortened by JJ, 26-Jul-2021)

		Ref	Expression
	Assertion	int0	⊢ ∩ ∅ = V

Proof

Step	Hyp	Ref	Expression
1		ral0	⊢ ∀ 𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex	⊢ 𝑦 ∈ V
3	2	elint2	⊢ ( 𝑦 ∈ ∩ ∅ ↔ ∀ 𝑥 ∈ ∅ 𝑦 ∈ 𝑥 )
4	1 3	mpbir	⊢ 𝑦 ∈ ∩ ∅
5	4 2	2th	⊢ ( 𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V )
6	5	eqriv	⊢ ∩ ∅ = V