Theorem nalfal 36942

Description: Not all sets hold ⊥ as true. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion

Ref	Expression
nalfal	⊢ ¬ ∀𝑥⊥

Proof of Theorem nalfal

Step	Hyp	Ref	Expression
1		alfal 1837	. 2 ⊢ ∀𝑥 ¬ ⊥
2		falim 1586	. . 3 ⊢ (⊥ → ¬ ∀𝑥 ¬ ⊥)
3	2	sps 2220	. 2 ⊢ (∀𝑥⊥ → ¬ ∀𝑥 ¬ ⊥)
4	1, 3	mt2 203	1 ⊢ ¬ ∀𝑥⊥

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567  ⊥wfal 1581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212

This proof depends on definitions:  df-bi 210  df-tru 1572  df-fal 1582  df-ex 1809

This theorem is used by: (None)