Description: A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013)
Ref | Expression | ||
---|---|---|---|
Hypothesis | snelsingles.1 | ⊢ 𝐴 ∈ V | |
Assertion | snelsingles | ⊢ { 𝐴 } ∈ Singletons |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelsingles.1 | ⊢ 𝐴 ∈ V | |
2 | isset | ⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) | |
3 | eqcom | ⊢ ( 𝑥 = 𝐴 ↔ 𝐴 = 𝑥 ) | |
4 | 3 | exbii | ⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑥 𝐴 = 𝑥 ) |
5 | 2 4 | bitri | ⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝐴 = 𝑥 ) |
6 | 1 5 | mpbi | ⊢ ∃ 𝑥 𝐴 = 𝑥 |
7 | sneq | ⊢ ( 𝐴 = 𝑥 → { 𝐴 } = { 𝑥 } ) | |
8 | 6 7 | eximii | ⊢ ∃ 𝑥 { 𝐴 } = { 𝑥 } |
9 | elsingles | ⊢ ( { 𝐴 } ∈ Singletons ↔ ∃ 𝑥 { 𝐴 } = { 𝑥 } ) | |
10 | 8 9 | mpbir | ⊢ { 𝐴 } ∈ Singletons |