Theorem zfnuleu 2712

Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 1465 to strengthen axnul 2714).

Hypothesis

Ref	Expression
zfnuleu.1	⊢ ∃x∀y ¬ y ∈ x

Assertion

Ref	Expression
zfnuleu	⊢ ∃!x∀y ¬ y ∈ x

Distinct variable group:   x,y

Proof of Theorem zfnuleu

Step	Hyp	Ref	Expression
1		zfnuleu.1	. . . 4 ⊢ ∃x∀y ¬ y ∈ x
2		equid 1128	. . . . . . 7 ⊢ y = y
3	2	nbn3 725	. . . . . 6 ⊢ (¬ y ∈ x ↔ (y ∈ x ↔ ¬ y = y))
4	3	albii 1001	. . . . 5 ⊢ (∀y ¬ y ∈ x ↔ ∀y(y ∈ x ↔ ¬ y = y))
5	4	exbii 1053	. . . 4 ⊢ (∃x∀y ¬ y ∈ x ↔ ∃x∀y(y ∈ x ↔ ¬ y = y))
6	1, 5	mpbi 189	. . 3 ⊢ ∃x∀y(y ∈ x ↔ ¬ y = y)
7		ax-17 973	. . . 4 ⊢ (¬ y = y → ∀x ¬ y = y)
8	7	bm1.1 1465	. . 3 ⊢ (∃x∀y(y ∈ x ↔ ¬ y = y) → ∃!x∀y(y ∈ x ↔ ¬ y = y))
9	6, 8	ax-mp 7	. 2 ⊢ ∃!x∀y(y ∈ x ↔ ¬ y = y)
10	4	eubii 1389	. 2 ⊢ (∃!x∀y ¬ y ∈ x ↔ ∃!x∀y(y ∈ x ↔ ¬ y = y))
11	9, 10	mpbir 190	1 ⊢ ∃!x∀y ¬ y ∈ x

Syntax hints:  ¬ wn 2   ↔ wb 146  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  ∃!weu 1382

This theorem is referenced by:  0ex 2716  snex 2756

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384

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