Theorem axnul 5224

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 5218. 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 2714).

This proof, suggested by Jeff Hoffman, uses only ax-4 1813 and ax-gen 1799 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 5218 implies the existence of at least one set. Note that Kunen's version of ax-sep 5218 (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 ∃𝑥𝑥 = 𝑥 (Axiom 0 of [Kunen] p. 10).

See axnulALT 5223 for a proof directly from ax-rep 5205.

This theorem should not be referenced by any proof. Instead, use ax-nul 5225 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	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem axnul

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 5218	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥))
2		fal 1553	. . . . 5 ⊢ ¬ ⊥
3	2	intnan 486	. . . 4 ⊢ ¬ (𝑦 ∈ 𝑧 ∧ ⊥)
4		id 22	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)))
5	3, 4	mtbiri 326	. . 3 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → ¬ 𝑦 ∈ 𝑥)
6	5	alimi 1815	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → ∀𝑦 ¬ 𝑦 ∈ 𝑥)
7	1, 6	eximii 1840	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395  ∀wal 1537  ⊥wfal 1551  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784

This theorem is referenced by: (None)