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Theorem axnul 4048
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 4041. 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 4047).

This theorem should not be referenced by any proof. Instead, use ax-nul 4049 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 4041 . 2 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦)))
2 pm3.24 682 . . . . . 6 ¬ (𝑦𝑦 ∧ ¬ 𝑦𝑦)
32intnan 914 . . . . 5 ¬ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦))
4 id 19 . . . . 5 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦))) → (𝑦𝑥 ↔ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦))))
53, 4mtbiri 664 . . . 4 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦))) → ¬ 𝑦𝑥)
65alimi 1431 . . 3 (∀𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦))) → ∀𝑦 ¬ 𝑦𝑥)
76eximi 1579 . 2 (∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ (𝑦𝑦 ∧ ¬ 𝑦𝑦))) → ∃𝑥𝑦 ¬ 𝑦𝑥)
81, 7ax-mp 5 1 𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wal 1329  wex 1468  wcel 1480
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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-sep 4041
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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