This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 0sdom1domALT

Description: Alternate proof of 0sdom1dom , shorter but requiring ax-un . (Contributed by NM, 28-Sep-2004) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 0sdom1domALT ( ∅ ≺ 𝐴 ↔ 1o ≼ 𝐴 )

Proof

Step Hyp Ref Expression
1 peano1 ⊢ ∅ ∈ ω
2 sucdom ⊢ ( ∅ ∈ ω → ( ∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴 ) )
3 1 2 ax-mp ⊢ ( ∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴 )
4 df-1o ⊢ 1o = suc ∅
5 4 breq1i ⊢ ( 1o ≼ 𝐴 ↔ suc ∅ ≼ 𝐴 )
6 3 5 bitr4i ⊢ ( ∅ ≺ 𝐴 ↔ 1o ≼ 𝐴 )