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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)))
2	1	simplbi 274	1 ⊢ (𝑅 TAp 𝐴 → 𝑅 Ap 𝐴)

Syntax hints:  ¬ wn 3   → wi 4  ∀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