Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > trunanfal | Structured version Visualization version GIF version |
Description: A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) |
Ref | Expression |
---|---|
trunanfal | ⊢ ((⊤ ⊼ ⊥) ↔ ⊤) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nan 1487 | . . 3 ⊢ ((⊤ ⊼ ⊥) ↔ ¬ (⊤ ∧ ⊥)) | |
2 | truanfal 1576 | . . 3 ⊢ ((⊤ ∧ ⊥) ↔ ⊥) | |
3 | 1, 2 | xchbinx 337 | . 2 ⊢ ((⊤ ⊼ ⊥) ↔ ¬ ⊥) |
4 | notfal 1570 | . 2 ⊢ (¬ ⊥ ↔ ⊤) | |
5 | 3, 4 | bitri 278 | 1 ⊢ ((⊤ ⊼ ⊥) ↔ ⊤) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ⊼ wnan 1486 ⊤wtru 1543 ⊥wfal 1554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-nan 1487 df-tru 1545 df-fal 1555 |
This theorem is referenced by: falnantru 1585 |
Copyright terms: Public domain | W3C validator |