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Theorem axnul 3910
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 3903. 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 3909).

This theorem should not be referenced by any proof. Instead, use ax-nul 3911 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  |-  E. x A. y  -.  y  e.  x
Distinct variable group:    x, y

Proof of Theorem axnul
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-sep 3903 . 2  |-  E. x A. y ( y  e.  x  <->  ( y  e.  z  /\  ( y  e.  y  /\  -.  y  e.  y )
) )
2 pm3.24 637 . . . . . 6  |-  -.  (
y  e.  y  /\  -.  y  e.  y
)
32intnan 849 . . . . 5  |-  -.  (
y  e.  z  /\  ( y  e.  y  /\  -.  y  e.  y ) )
4 id 19 . . . . 5  |-  ( ( y  e.  x  <->  ( y  e.  z  /\  (
y  e.  y  /\  -.  y  e.  y
) ) )  -> 
( y  e.  x  <->  ( y  e.  z  /\  ( y  e.  y  /\  -.  y  e.  y ) ) ) )
53, 4mtbiri 610 . . . 4  |-  ( ( y  e.  x  <->  ( y  e.  z  /\  (
y  e.  y  /\  -.  y  e.  y
) ) )  ->  -.  y  e.  x
)
65alimi 1360 . . 3  |-  ( A. y ( y  e.  x  <->  ( y  e.  z  /\  ( y  e.  y  /\  -.  y  e.  y )
) )  ->  A. y  -.  y  e.  x
)
76eximi 1507 . 2  |-  ( E. x A. y ( y  e.  x  <->  ( y  e.  z  /\  (
y  e.  y  /\  -.  y  e.  y
) ) )  ->  E. x A. y  -.  y  e.  x )
81, 7ax-mp 7 1  |-  E. x A. y  -.  y  e.  x
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    <-> wb 102   A.wal 1257   E.wex 1397    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-sep 3903
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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