Metamath Proof Explorer


Theorem snct

Description: A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016)

Ref Expression
Assertion snct ( 𝐴𝑉 → { 𝐴 } ≼ ω )

Proof

Step Hyp Ref Expression
1 ensn1g ( 𝐴𝑉 → { 𝐴 } ≈ 1o )
2 peano1 ∅ ∈ ω
3 2 ne0ii ω ≠ ∅
4 omex ω ∈ V
5 4 0sdom ( ∅ ≺ ω ↔ ω ≠ ∅ )
6 3 5 mpbir ∅ ≺ ω
7 0sdom1dom ( ∅ ≺ ω ↔ 1o ≼ ω )
8 6 7 mpbi 1o ≼ ω
9 endomtr ( ( { 𝐴 } ≈ 1o ∧ 1o ≼ ω ) → { 𝐴 } ≼ ω )
10 1 8 9 sylancl ( 𝐴𝑉 → { 𝐴 } ≼ ω )