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)

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

Proof

Step	Hyp	Ref	Expression
1		sucdom2	⊢ ( 𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵 )
2		php4	⊢ ( 𝐴 ∈ ω → 𝐴 ≺ suc 𝐴 )
3		sdomdomtr	⊢ ( ( 𝐴 ≺ suc 𝐴 ∧ suc 𝐴 ≼ 𝐵 ) → 𝐴 ≺ 𝐵 )
4	3	ex	⊢ ( 𝐴 ≺ suc 𝐴 → ( suc 𝐴 ≼ 𝐵 → 𝐴 ≺ 𝐵 ) )
5	2 4	syl	⊢ ( 𝐴 ∈ ω → ( suc 𝐴 ≼ 𝐵 → 𝐴 ≺ 𝐵 ) )
6	1 5	impbid2	⊢ ( 𝐴 ∈ ω → ( 𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵 ) )