Theorem axnul 3882

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 3875. 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 3881).

This theorem should not be referenced by any proof. Instead, use ax-nul 3883 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 3875	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦)))
2		pm3.24 627	. . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦)
3	2	intnan 838	. . . . 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 600	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → ¬ 𝑦 ∈ 𝑥)
6	5	alimi 1344	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → ∀𝑦 ¬ 𝑦 ∈ 𝑥)
7	6	eximi 1491	. 2 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ (𝑦 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑦))) → ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
8	1, 7	ax-mp 7	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-sep 3875

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)