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Metamath Proof Explorer

Theorem dfnul2

Description: Alternate definition of the empty set. Definition 5.14 of TakeutiZaring p. 20. (Contributed by NM, 26-Dec-1996) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024)

Ref Expression
Assertion dfnul2 
|- (/) = { x | -. x = x }

Proof

Step Hyp Ref Expression
1 dfnul4 
 |-  (/) = { x | F. }
2 equid 
 |-  x = x
3 2 notnoti 
 |-  -. -. x = x
4 3 bifal 
 |-  ( -. x = x <-> F. )
5 4 abbii 
 |-  { x | -. x = x } = { x | F. }
6 1 5 eqtr4i 
 |-  (/) = { x | -. x = x }