Theorem uniintab

Description: The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of ph ( x ) . (Contributed by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Assertion	uniintab	⊢ ( ∃! 𝑥 𝜑 ↔ ∪ { 𝑥 ∣ 𝜑 } = ∩ { 𝑥 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		euabsn2	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } )
2		uniintsn	⊢ ( ∪ { 𝑥 ∣ 𝜑 } = ∩ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑦 { 𝑥 ∣ 𝜑 } = { 𝑦 } )
3	1 2	bitr4i	⊢ ( ∃! 𝑥 𝜑 ↔ ∪ { 𝑥 ∣ 𝜑 } = ∩ { 𝑥 ∣ 𝜑 } )