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Metamath Proof Explorer

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 ) )