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 | ⊢ ∃ 𝑥 ∀ 𝑦 ¬ 𝑦 ∈ 𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-sep | ⊢ ∃ 𝑥 ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝑧 ∧ ⊥ ) ) | |
| 2 | fal | ⊢ ¬ ⊥ | |
| 3 | 2 | intnan | ⊢ ¬ ( 𝑦 ∈ 𝑧 ∧ ⊥ ) |
| 4 | id | ⊢ ( ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝑧 ∧ ⊥ ) ) → ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝑧 ∧ ⊥ ) ) ) | |
| 5 | 3 4 | mtbiri | ⊢ ( ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝑧 ∧ ⊥ ) ) → ¬ 𝑦 ∈ 𝑥 ) |
| 6 | 5 | alimi | ⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝑧 ∧ ⊥ ) ) → ∀ 𝑦 ¬ 𝑦 ∈ 𝑥 ) |
| 7 | 1 6 | eximii | ⊢ ∃ 𝑥 ∀ 𝑦 ¬ 𝑦 ∈ 𝑥 |