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Metamath Proof Explorer

Theorem bnj158

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypothesis bnj158.1 ⊢ D = ω ∖ ∅
Assertion bnj158 ⊢ m ∈ D → ∃ p ∈ ω m = suc ⁡ p

Proof

Step Hyp Ref Expression
1 bnj158.1 ⊢ D = ω ∖ ∅
2 1 eleq2i ⊢ m ∈ D ↔ m ∈ ω ∖ ∅
3 eldifsn ⊢ m ∈ ω ∖ ∅ ↔ m ∈ ω ∧ m ≠ ∅
4 2 3 bitri ⊢ m ∈ D ↔ m ∈ ω ∧ m ≠ ∅
5 nnsuc ⊢ m ∈ ω ∧ m ≠ ∅ → ∃ p ∈ ω m = suc ⁡ p
6 4 5 sylbi ⊢ m ∈ D → ∃ p ∈ ω m = suc ⁡ p