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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 ∈ Fin → 𝐴 ≈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		f1oi	⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴
2		f1oenfi	⊢ ( ( 𝐴 ∈ Fin ∧ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 ) → 𝐴 ≈ 𝐴 )
3	1 2	mpan2	⊢ ( 𝐴 ∈ Fin → 𝐴 ≈ 𝐴 )