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Theorem empty 1907
Description: Two characterizations of the empty domain. (Contributed by GΓ©rard Lang, 5-Feb-2024.)
Assertion
Ref Expression
empty (Β¬ βˆƒπ‘₯⊀ ↔ βˆ€π‘₯βŠ₯)

Proof of Theorem empty
StepHypRef Expression
1 df-fal 1552 . . 3 (βŠ₯ ↔ Β¬ ⊀)
21albii 1819 . 2 (βˆ€π‘₯βŠ₯ ↔ βˆ€π‘₯ Β¬ ⊀)
3 alnex 1781 . 2 (βˆ€π‘₯ Β¬ ⊀ ↔ Β¬ βˆƒπ‘₯⊀)
42, 3bitr2i 275 1 (Β¬ βˆƒπ‘₯⊀ ↔ βˆ€π‘₯βŠ₯)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205  βˆ€wal 1537  βŠ€wtru 1540  βŠ₯wfal 1551  βˆƒwex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809
This theorem depends on definitions:  df-bi 206  df-fal 1552  df-ex 1780
This theorem is referenced by: (None)
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