Theorem enreffi

Description: Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg ). (Contributed by BTernaryTau, 8-Sep-2024)

	Ref	Expression
Assertion	enreffi	|- ( A e. Fin -> A ~~ A )

Proof

Step	Hyp	Ref	Expression
1		f1oi	|- ( _I |` A ) : A -1-1-onto-> A
2		f1oenfi	|- ( ( A e. Fin /\ ( _I |` A ) : A -1-1-onto-> A ) -> A ~~ A )
3	1 2	mpan2	|- ( A e. Fin -> A ~~ A )