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Mirrors > Home > ILE Home > Th. List > axnul | Unicode version |
Description: The Null Set Axiom of ZF
set theory: there exists a set with no
elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks,
this is presented as a separate axiom; here we show it can be derived
from Separation ax-sep 4054. This version of the Null Set Axiom tells us
that at least one empty set exists, but does not tell us that it is
unique - we need the Axiom of Extensionality to do that (see
zfnuleu 4060).
This theorem should not be referenced by any proof. Instead, use ax-nul 4062 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
axnul |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-sep 4054 |
. 2
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2 | pm3.24 683 |
. . . . . 6
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3 | 2 | intnan 915 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | id 19 |
. . . . 5
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5 | 3, 4 | mtbiri 665 |
. . . 4
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6 | 5 | alimi 1432 |
. . 3
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7 | 6 | eximi 1580 |
. 2
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8 | 1, 7 | ax-mp 5 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 ax-sep 4054 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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