Theorem dfin5

Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005)

	Ref	Expression
Assertion	dfin5	⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 }

Proof

Step	Hyp	Ref	Expression
1		df-in	⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) }
2		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) }
3	1 2	eqtr4i	⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 }