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Theorem axnul 3856
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 3849. 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 3855).

This theorem should not be referenced by any proof. Instead, use ax-nul 3857 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 ¬ y x
Distinct variable group:   x,y

Proof of Theorem axnul
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ax-sep 3849 . 2 xy(y x ↔ (y z (y y ¬ y y)))
2 pm3.24 614 . . . . . 6 ¬ (y y ¬ y y)
32intnan 826 . . . . 5 ¬ (y z (y y ¬ y y))
4 id 19 . . . . 5 ((y x ↔ (y z (y y ¬ y y))) → (y x ↔ (y z (y y ¬ y y))))
53, 4mtbiri 587 . . . 4 ((y x ↔ (y z (y y ¬ y y))) → ¬ y x)
65alimi 1324 . . 3 (y(y x ↔ (y z (y y ¬ y y))) → y ¬ y x)
76eximi 1473 . 2 (xy(y x ↔ (y z (y y ¬ y y))) → xy ¬ y x)
81, 7ax-mp 7 1 xy ¬ y x
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98  wal 1226  wex 1362   wcel 1374
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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409  ax-sep 3849
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
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