Theorem axnul 2705

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 2699. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tells us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 2703).

This proof, suggested by Jeff Hoffman (3-Feb-2008), does not require the set existence axiom ax-9 964, unlike some empty set existence proofs (such as the remark in [Kunen] p. 11). However, it uses ax-4 972, which also presupposes the existence of at least one set, i.e it does not hold in a "free logic" valid in empty domains. Theorem ax4 971, which shows the redundancy of ax-4 972, depends on ax-9 964 for its proof. See axnul2 2704 for a similar proof directly from ax-rep 2689.

This theorem should not be referenced by any proof. Instead, use ax-nul 2706 below so that the uses of the Null Set Axiom can be more easily identified.

Assertion

Ref	Expression
axnul	⊢ ∃x∀y ¬ y ∈ x

Distinct variable group:   x,y

Proof of Theorem axnul

Step	Hyp	Ref	Expression
1		ax-sep 2699	. 2 ⊢ ∃x∀y(y ∈ x ↔ (y ∈ z ⋀ (y ∈ y ⋀ ¬ y ∈ y)))
2		pm3.24 657	. . . . . 6 ⊢ ¬ (y ∈ y ⋀ ¬ y ∈ y)
3	2	intnan 690	. . . . 5 ⊢ ¬ (y ∈ z ⋀ (y ∈ y ⋀ ¬ y ∈ y))
4		id 59	. . . . 5 ⊢ ((y ∈ x ↔ (y ∈ z ⋀ (y ∈ y ⋀ ¬ y ∈ y))) → (y ∈ x ↔ (y ∈ z ⋀ (y ∈ y ⋀ ¬ y ∈ y))))
5	3, 4	mtbiri 716	. . . 4 ⊢ ((y ∈ x ↔ (y ∈ z ⋀ (y ∈ y ⋀ ¬ y ∈ y))) → ¬ y ∈ x)
6	5	19.20i 991	. . 3 ⊢ (∀y(y ∈ x ↔ (y ∈ z ⋀ (y ∈ y ⋀ ¬ y ∈ y))) → ∀y ¬ y ∈ x)
7	6	19.22i 1039	. 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 2   ↔ wb 146   ⋀ wa 223  ∀wal 953   ∈ wcel 957  ∃wex 979

This theorem is referenced by:  snex 2746

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-4 972  ax-5o 974  ax-sep 2699

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980

Copyright terms: Public domain