# Metamath Proof Explorer

## Theorem axnul

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 . 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 nulmo ).

This proof, suggested by Jeff Hoffman, uses only ax-4 and ax-gen 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 implies the existence of at least one set. Note that Kunen's version of ax-sep (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 for a proof directly from ax-rep .

This theorem should not be referenced by any proof. Instead, use ax-nul 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.)

Ref Expression
Assertion axnul ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\forall {y}\phantom{\rule{.4em}{0ex}}¬{y}\in {x}$

### Proof

Step Hyp Ref Expression
1 ax-sep ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}\in {x}↔\left({y}\in {z}\wedge \perp \right)\right)$
2 fal ${⊢}¬\perp$
3 2 intnan ${⊢}¬\left({y}\in {z}\wedge \perp \right)$
4 id ${⊢}\left({y}\in {x}↔\left({y}\in {z}\wedge \perp \right)\right)\to \left({y}\in {x}↔\left({y}\in {z}\wedge \perp \right)\right)$
5 3 4 mtbiri ${⊢}\left({y}\in {x}↔\left({y}\in {z}\wedge \perp \right)\right)\to ¬{y}\in {x}$
6 5 alimi ${⊢}\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}\in {x}↔\left({y}\in {z}\wedge \perp \right)\right)\to \forall {y}\phantom{\rule{.4em}{0ex}}¬{y}\in {x}$
7 1 6 eximii ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\forall {y}\phantom{\rule{.4em}{0ex}}¬{y}\in {x}$