Theorem kardfi

Description: The kard cardinal number of a finite set is finite. (Contributed by BTernaryTau, 3-Jul-2026)

	Ref	Expression
Assertion	kardfi	⊢ ( 𝐴 ∈ Fin → ( kard ‘ 𝐴 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		isfi	⊢ ( 𝐴 ∈ Fin ↔ ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 )
2		df-rex	⊢ ( ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥 ) )
3		kardeng	⊢ ( 𝐴 ∈ Fin → ( ( kard ‘ 𝐴 ) = ( kard ‘ 𝑥 ) ↔ 𝐴 ≈ 𝑥 ) )
4	3	biimprd	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ≈ 𝑥 → ( kard ‘ 𝐴 ) = ( kard ‘ 𝑥 ) ) )
5		kardnnfi	⊢ ( 𝑥 ∈ ω → ( kard ‘ 𝑥 ) ∈ Fin )
6		eleq1a	⊢ ( ( kard ‘ 𝑥 ) ∈ Fin → ( ( kard ‘ 𝐴 ) = ( kard ‘ 𝑥 ) → ( kard ‘ 𝐴 ) ∈ Fin ) )
7	5 6	syl	⊢ ( 𝑥 ∈ ω → ( ( kard ‘ 𝐴 ) = ( kard ‘ 𝑥 ) → ( kard ‘ 𝐴 ) ∈ Fin ) )
8	7	imp	⊢ ( ( 𝑥 ∈ ω ∧ ( kard ‘ 𝐴 ) = ( kard ‘ 𝑥 ) ) → ( kard ‘ 𝐴 ) ∈ Fin )
9	8	a1i	⊢ ( 𝐴 ∈ Fin → ( ( 𝑥 ∈ ω ∧ ( kard ‘ 𝐴 ) = ( kard ‘ 𝑥 ) ) → ( kard ‘ 𝐴 ) ∈ Fin ) )
10	4 9	sylan2d	⊢ ( 𝐴 ∈ Fin → ( ( 𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥 ) → ( kard ‘ 𝐴 ) ∈ Fin ) )
11	10	exlimdv	⊢ ( 𝐴 ∈ Fin → ( ∃ 𝑥 ( 𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥 ) → ( kard ‘ 𝐴 ) ∈ Fin ) )
12	2 11	biimtrid	⊢ ( 𝐴 ∈ Fin → ( ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 → ( kard ‘ 𝐴 ) ∈ Fin ) )
13	1 12	biimtrid	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∈ Fin → ( kard ‘ 𝐴 ) ∈ Fin ) )
14	13	pm2.43i	⊢ ( 𝐴 ∈ Fin → ( kard ‘ 𝐴 ) ∈ Fin )