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Theorem axnul 4297
 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 4290. 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 4295). This proof, suggested by Jeff Hoffman, uses only ax-5 1563 and ax-gen 1552 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus, our ax-sep 4290 implies the existence of at least one set. Note that Kunen's version of ax-sep 4290 (Axiom 3 of [Kunen] p. 11) does not imply the existence of a set because his is universally closed i.e. prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating (Axiom 0 of [Kunen] p. 10). See axnulALT 4296 for a proof directly from ax-rep 4280. This theorem should not be referenced by any proof. Instead, use ax-nul 4298 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.)
Assertion
Ref Expression
axnul
Distinct variable group:   ,

Proof of Theorem axnul
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4290 . 2
2 pm3.24 853 . . . . . 6
32intnan 881 . . . . 5
4 id 20 . . . . 5
53, 4mtbiri 295 . . . 4
65alimi 1565 . . 3
76eximi 1582 . 2
81, 7ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wa 359  wal 1546  wex 1547   wcel 1721 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-sep 4290 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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