Description: A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | snct | ⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ≼ ω ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensn1g | ⊢ ( 𝐴 ∈ 𝑉 → { 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 | ⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ≼ ω ) |