Metamath Proof Explorer


Theorem snelsingles

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

Proof

Step Hyp Ref Expression
1 snelsingles.1 𝐴 ∈ V
2 isset ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
3 eqcom ( 𝑥 = 𝐴𝐴 = 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