Theorem axnul 4107

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 4100. 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 4106).

This theorem should not be referenced by any proof. Instead, use ax-nul 4108 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 4100	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦)))
2		pm3.24 683	. . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦)
3	2	intnan 919	. . . . 5 ⊢ ¬ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))
4		id 19	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))))
5	3, 4	mtbiri 665	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → ¬ 𝑦 ∈ 𝑥)
6	5	alimi 1443	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → ∀𝑦 ¬ 𝑦 ∈ 𝑥)
7	6	eximi 1588	. 2 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
8	1, 7	ax-mp 5	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104  ∀wal 1341  ∃wex 1480   ∈ wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-sep 4100

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)