Metamath Proof Explorer

Definition df-eu

Description: Define the existential uniqueness quantifier. This expresses unique existence, or existential uniqueness, which is the conjunction of existence ( df-ex ) and uniqueness ( df-mo ). The expression E! x ph is read "there exists exactly one x such that ph " or "there exists a unique x such that ph ". This is also called the "uniqueness quantifier" but that expression is also used for the at-most-one quantifier df-mo , therefore we avoid that ambiguous name.

Definition 10.1 of BellMachover p. 97; also Definition *14.02 of WhiteheadRussell p. 175. Other possible definitions are given by eu1 , eu2 , eu3v , and eu6 . As for double unique existence, beware that the expression E! x E! y ph means "there exists a unique x such that there exists a unique y such that ph " which is a weaker property than "there exists exactly one x and one y such that ph " (see 2eu4 ). (Contributed by NM, 12-Aug-1993) Make this the definition (which used to be eu6 , while this definition was then proved as dfeu ). (Revised by BJ, 30-Sep-2022)

Ref Expression
Assertion df-eu
`|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )`

Detailed syntax breakdown

Step Hyp Ref Expression
0 vx
` |-  x`
1 wph
` |-  ph`
2 1 0 weu
` |-  E! x ph`
3 1 0 wex
` |-  E. x ph`
4 1 0 wmo
` |-  E* x ph`
5 3 4 wa
` |-  ( E. x ph /\ E* x ph )`
6 2 5 wb
` |-  ( E! x ph <-> ( E. x ph /\ E* x ph ) )`