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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality The difference, union, and intersection of two classes The intersection of two classes incom
Description: Commutative law for intersection of classes. Exercise 7 of
TakeutiZaring p. 17. (Contributed by NM , 21-Jun-1993) (Proof
shortened by SN , 12-Dec-2023)
Ref
Expression
Assertion
incom ⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
Proof
Step
Hyp
Ref
Expression
1
rabswap ⊢ { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 } = { 𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴 }
2
dfin5 ⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 }
3
dfin5 ⊢ ( 𝐵 ∩ 𝐴 ) = { 𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴 }
4
1 2 3
3eqtr4i ⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )