Class Equality is Symmetric
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Theorem
Let $A$ and $B$ be classes.
Let $=$ denote class equality.
Then:
- $A = B \implies B = A$
Proof
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From Biconditional is Commutative:
- $\forall x: \left({ x \in A \iff x \in B }\right) \implies \forall x: \left({ x \in B \iff x \in A }\right)$
Hence the result by definition of class equality.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 4.7 \ (2)$