Theorem bdne 16174

Description: Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdne	⊢ BOUNDED 𝑥 ≠ 𝑦

Proof of Theorem bdne

Step	Hyp	Ref	Expression
1		ax-bdeq 16141	. . 3 ⊢ BOUNDED 𝑥 = 𝑦
2	1	ax-bdn 16138	. 2 ⊢ BOUNDED ¬ 𝑥 = 𝑦
3		df-ne 2401	. 2 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
4	2, 3	bd0r 16146	1 ⊢ BOUNDED 𝑥 ≠ 𝑦

Syntax hints:  ¬ wn 3   ≠ wne 2400  BOUNDED wbd 16133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-bd0 16134  ax-bdn 16138  ax-bdeq 16141

This theorem depends on definitions:  df-bi 117  df-ne 2401

This theorem is referenced by: (None)