Metamath Proof Explorer


Theorem sniota

Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016)

Ref Expression
Assertion sniota ∃! x φ x | φ = ι x | φ

Proof

Step Hyp Ref Expression
1 nfeu1 x ∃! x φ
2 nfab1 _ x x | φ
3 nfiota1 _ x ι x | φ
4 3 nfsn _ x ι x | φ
5 iota1 ∃! x φ φ ι x | φ = x
6 eqcom ι x | φ = x x = ι x | φ
7 5 6 syl6bb ∃! x φ φ x = ι x | φ
8 abid x x | φ φ
9 velsn x ι x | φ x = ι x | φ
10 7 8 9 3bitr4g ∃! x φ x x | φ x ι x | φ
11 1 2 4 10 eqrd ∃! x φ x | φ = ι x | φ