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Metamath Proof Explorer

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	$⊢ A \in Fin \to A \approx A$

Proof

Step	Hyp	Ref	Expression
1		f1oi	$⊢ ({I ↾}_{A}) : A \underset{1-1 onto}{⟶} A$
2		f1oenfi	$⊢ (A \in Fin \land ({I ↾}_{A}) : A \underset{1-1 onto}{⟶} A) \to A \approx A$
3	1 2	mpan2	$⊢ A \in Fin \to A \approx A$