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Metamath Proof Explorer

Theorem difrab0eq

Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017)

		Ref	Expression
	Assertion	difrab0eq	⊢ ( ( 𝑉 ∖ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ) = ∅ ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		ssdif0	⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ( 𝑉 ∖ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ) = ∅ )
2		ssrabeq	⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )
3	1 2	bitr3i	⊢ ( ( 𝑉 ∖ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ) = ∅ ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )