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Metamath Proof Explorer

Theorem ab0orvALT

Description: Alternate proof of ab0orv , shorter but using more axioms. (Contributed by Mario Carneiro, 29-Aug-2013) (Revised by BJ, 22-Mar-2020) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	ab0orvALT	⊢ ( { 𝑥 ∣ 𝜑 } = V ∨ { 𝑥 ∣ 𝜑 } = ∅ )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑥 𝜑
2		dfnf5	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( { 𝑥 ∣ 𝜑 } = V ∨ { 𝑥 ∣ 𝜑 } = ∅ ) )
3	1 2	mpbi	⊢ ( { 𝑥 ∣ 𝜑 } = V ∨ { 𝑥 ∣ 𝜑 } = ∅ )