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Mirrors > Home > MPE Home > Th. List > empty | Structured version Visualization version GIF version |
Description: Two characterizations of the empty domain. (Contributed by GΓ©rard Lang, 5-Feb-2024.) |
Ref | Expression |
---|---|
empty | β’ (Β¬ βπ₯β€ β βπ₯β₯) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fal 1552 | . . 3 β’ (β₯ β Β¬ β€) | |
2 | 1 | albii 1819 | . 2 β’ (βπ₯β₯ β βπ₯ Β¬ β€) |
3 | alnex 1781 | . 2 β’ (βπ₯ Β¬ β€ β Β¬ βπ₯β€) | |
4 | 2, 3 | bitr2i 275 | 1 β’ (Β¬ βπ₯β€ β βπ₯β₯) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 βwal 1537 β€wtru 1540 β₯wfal 1551 βwex 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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