Theorem 0ex

Description: The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of TakeutiZaring p. 20. For the unabbreviated version, see ax-nul . (Contributed by NM, 21-Jun-1993) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	0ex	⊢ ∅ ∈ V

Proof

Step	Hyp	Ref	Expression
1		ax-nul	⊢ ∃ 𝑥 ∀ 𝑦 ¬ 𝑦 ∈ 𝑥
2		eq0	⊢ ( 𝑥 = ∅ ↔ ∀ 𝑦 ¬ 𝑦 ∈ 𝑥 )
3	2	exbii	⊢ ( ∃ 𝑥 𝑥 = ∅ ↔ ∃ 𝑥 ∀ 𝑦 ¬ 𝑦 ∈ 𝑥 )
4	1 3	mpbir	⊢ ∃ 𝑥 𝑥 = ∅
5	4	issetri	⊢ ∅ ∈ V