Theorem bdne 13735

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 13702	. . 3 ⊢ BOUNDED 𝑥 = 𝑦
2	1	ax-bdn 13699	. 2 ⊢ BOUNDED ¬ 𝑥 = 𝑦
3		df-ne 2337	. 2 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
4	2, 3	bd0r 13707	1 ⊢ BOUNDED 𝑥 ≠ 𝑦

Syntax hints:  ¬ wn 3   ≠ wne 2336  BOUNDED wbd 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-bd0 13695  ax-bdn 13699  ax-bdeq 13702

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by: (None)