Metamath Proof Explorer


Theorem 1sno

Description: Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024)

Ref Expression
Assertion 1sno 1s ∈ No

Proof

Step Hyp Ref Expression
1 df-1s 1s = ( { 0s } |s ∅ )
2 0sno 0s ∈ No
3 snelpwi ( 0s ∈ No → { 0s } ∈ 𝒫 No )
4 2 3 ax-mp { 0s } ∈ 𝒫 No
5 nulssgt ( { 0s } ∈ 𝒫 No → { 0s } <<s ∅ )
6 4 5 ax-mp { 0s } <<s ∅
7 scutcl ( { 0s } <<s ∅ → ( { 0s } |s ∅ ) ∈ No )
8 6 7 ax-mp ( { 0s } |s ∅ ) ∈ No
9 1 8 eqeltri 1s ∈ No