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Theorem axnul 4164
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 4157. 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 4162).

This proof, suggested by Jeff Hoffman, uses only ax-5 1547 and ax-gen 1536 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 4157 implies the existence of at least one set. Note that Kunen's version of ax-sep 4157 (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  E. x x  =  x (Axiom 0 of [Kunen] p. 10).

See axnulALT 4163 for a proof directly from ax-rep 4147.

This theorem should not be referenced by any proof. Instead, use ax-nul 4165 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 4157 . 2  |-  E. x A. y ( y  e.  x  <->  ( y  e.  z  /\  ( y  e.  y  /\  -.  y  e.  y )
) )
2 pm3.24 852 . . . . . 6  |-  -.  (
y  e.  y  /\  -.  y  e.  y
)
32intnan 880 . . . . 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 294 . . . 4  |-  ( ( y  e.  x  <->  ( y  e.  z  /\  (
y  e.  y  /\  -.  y  e.  y
) ) )  ->  -.  y  e.  x
)
65alimi 1549 . . 3  |-  ( A. y ( y  e.  x  <->  ( y  e.  z  /\  ( y  e.  y  /\  -.  y  e.  y )
) )  ->  A. y  -.  y  e.  x
)
76eximi 1566 . 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 8 1  |-  E. x A. y  -.  y  e.  x
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    e. wcel 1696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator