Theorem nnfi

Description: Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015) Avoid ax-pow . (Revised by BTernaryTau, 23-Sep-2024)

	Ref	Expression
Assertion	nnfi	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		enrefnn	⊢ ( 𝐴 ∈ ω → 𝐴 ≈ 𝐴 )
2		breq2	⊢ ( 𝑥 = 𝐴 → ( 𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴 ) )
3	2	rspcev	⊢ ( ( 𝐴 ∈ ω ∧ 𝐴 ≈ 𝐴 ) → ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 )
4	1 3	mpdan	⊢ ( 𝐴 ∈ ω → ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 )
5		isfi	⊢ ( 𝐴 ∈ Fin ↔ ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 )
6	4 5	sylibr	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin )