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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 3866. 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 3872).
This theorem should not be referenced by any proof. Instead, use ax-nul 3874 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 3866 |
. 2
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2 | pm3.24 626 |
. . . . . 6
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3 | 2 | intnan 837 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | id 19 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | mtbiri 599 |
. . . 4
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6 | 5 | alimi 1341 |
. . 3
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7 | 6 | eximi 1488 |
. 2
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8 | 1, 7 | ax-mp 7 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 ax-sep 3866 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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