Theorem tapap 7569

Description: A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)

Assertion

Ref	Expression
tapap	|- ( R TAp A -> R Ap A )

Proof of Theorem tapap

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tap 7568	. 2 |- ( R TAp A <-> ( R Ap A /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) )
2	1	simplbi 274	1 |- ( R TAp A -> R Ap A )

Syntax hints:   -. wn 3   -> wi 4   A.wral 2522   class class class wbr 4111   Ap wap 7560   TAp wtap 7567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-tap 7568

This theorem is referenced by:  drnglring  14467