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Metamath Proof Explorer

Theorem eq0rdvALT

Description: Alternate proof of eq0rdv . Shorter, but requiring df-clel , ax-8 . (Contributed by NM, 11-Jul-2014) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	eq0rdvALT.1	⊢ ( 𝜑 → ¬ 𝑥 ∈ 𝐴 )
	Assertion	eq0rdvALT	⊢ ( 𝜑 → 𝐴 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		eq0rdvALT.1	⊢ ( 𝜑 → ¬ 𝑥 ∈ 𝐴 )
2	1	pm2.21d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ∅ ) )
3	2	ssrdv	⊢ ( 𝜑 → 𝐴 ⊆ ∅ )
4		ss0	⊢ ( 𝐴 ⊆ ∅ → 𝐴 = ∅ )
5	3 4	syl	⊢ ( 𝜑 → 𝐴 = ∅ )