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Theorem axnul 4923
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 4916. 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 4921).

This proof, suggested by Jeff Hoffman, uses only ax-4 1885 and ax-gen 1870 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 4916 implies the existence of at least one set. Note that Kunen's version of ax-sep 4916 (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 𝑥𝑥 = 𝑥 (Axiom 0 of [Kunen] p. 10).

See axnulALT 4922 for a proof directly from ax-rep 4905.

This theorem should not be referenced by any proof. Instead, use ax-nul 4924 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 𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem axnul
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4916 . 2 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥))
2 fal 1638 . . . . 5 ¬ ⊥
32intnan 996 . . . 4 ¬ (𝑦𝑧 ∧ ⊥)
4 id 22 . . . 4 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)))
53, 4mtbiri 316 . . 3 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → ¬ 𝑦𝑥)
65alimi 1887 . 2 (∀𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → ∀𝑦 ¬ 𝑦𝑥)
71, 6eximii 1912 1 𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wal 1629  wfal 1636  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-sep 4916
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-fal 1637  df-ex 1853
This theorem is referenced by: (None)
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