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Metamath Proof Explorer

Theorem eq0ALT

Description: Alternate proof of eq0 . Shorter, but requiring df-clel , ax-8 . (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	eq0ALT	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅ ) )
2		noel	⊢ ¬ 𝑥 ∈ ∅
3	2	nbn	⊢ ( ¬ 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅ ) )
4	3	albii	⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅ ) )
5	1 4	bitr4i	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )