Description: Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009) (Revised by Mario Carneiro, 22-Dec-2013) (Revised by AV, 18-Aug-2026)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | mgmideud.e | |- ( ph -> E. u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) ) |
|
| Assertion | mgmideud | |- ( ph -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mgmideud.e | |- ( ph -> E. u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) ) |
|
| 2 | mgmidmo | |- E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) |
|
| 3 | reu5 | |- ( E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( E. u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) ) ) |
|
| 4 | 1 2 3 | sylanblrc | |- ( ph -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) ) |