Theorem notab

Description: A class abstraction defined by a negation. (Contributed by FL, 18-Sep-2010)

	Ref	Expression
Assertion	notab	⊢ { 𝑥 ∣ ¬ 𝜑 } = ( V ∖ { 𝑥 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		df-rab	⊢ { 𝑥 ∈ V ∣ ¬ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝜑 ) }
2		rabab	⊢ { 𝑥 ∈ V ∣ ¬ 𝜑 } = { 𝑥 ∣ ¬ 𝜑 }
3	1 2	eqtr3i	⊢ { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝜑 ) } = { 𝑥 ∣ ¬ 𝜑 }
4		difab	⊢ ( { 𝑥 ∣ 𝑥 ∈ V } ∖ { 𝑥 ∣ 𝜑 } ) = { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝜑 ) }
5		abid2	⊢ { 𝑥 ∣ 𝑥 ∈ V } = V
6	5	difeq1i	⊢ ( { 𝑥 ∣ 𝑥 ∈ V } ∖ { 𝑥 ∣ 𝜑 } ) = ( V ∖ { 𝑥 ∣ 𝜑 } )
7	4 6	eqtr3i	⊢ { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝜑 ) } = ( V ∖ { 𝑥 ∣ 𝜑 } )
8	3 7	eqtr3i	⊢ { 𝑥 ∣ ¬ 𝜑 } = ( V ∖ { 𝑥 ∣ 𝜑 } )