Theorem sucdom

Description: Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013) Avoid ax-pow . (Revised by BTernaryTau, 4-Dec-2024) (Proof shortened by BJ, 11-Jan-2025)

	Ref	Expression
Assertion	sucdom	⊢ ( 𝐴 ∈ ω → ( 𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sucdom2	⊢ ( 𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵 )
2		nnfi	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin )
3		php4	⊢ ( 𝐴 ∈ ω → 𝐴 ≺ suc 𝐴 )
4		sdomdomtrfi	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ≺ suc 𝐴 ∧ suc 𝐴 ≼ 𝐵 ) → 𝐴 ≺ 𝐵 )
5	4	3expia	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ≺ suc 𝐴 ) → ( suc 𝐴 ≼ 𝐵 → 𝐴 ≺ 𝐵 ) )
6	2 3 5	syl2anc	⊢ ( 𝐴 ∈ ω → ( suc 𝐴 ≼ 𝐵 → 𝐴 ≺ 𝐵 ) )
7	1 6	impbid2	⊢ ( 𝐴 ∈ ω → ( 𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵 ) )