Theorem trunanfal 1355

Description: A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Assertion

Ref	Expression
trunanfal	⊢ (( ⊤ ⊼ ⊥ ) ↔ ⊤ )

Proof of Theorem trunanfal

Step	Hyp	Ref	Expression
1		df-nan 1288	. 2 ⊢ (( ⊤ ⊼ ⊥ ) ↔ ¬ ( ⊤ ∧ ⊥ ))
2		truanfal 1337	. . 3 ⊢ (( ⊤ ∧ ⊥ ) ↔ ⊥ )
3	2	notbii 287	. 2 ⊢ (¬ ( ⊤ ∧ ⊥ ) ↔ ¬ ⊥ )
4		notfal 1349	. 2 ⊢ (¬ ⊥ ↔ ⊤ )
5	1, 3, 4	3bitri 262	1 ⊢ (( ⊤ ⊼ ⊥ ) ↔ ⊤ )

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287   ⊤ wtru 1316   ⊥ wfal 1317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-nan 1288  df-tru 1319  df-fal 1320

This theorem is used by:  falnantru  1356