Theorem zfnuleu 2702

Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 1460 to strengthen axnul 2704).

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 1124	. . . . . . 7 ⊢ y = y
3	2	nbn3 722	. . . . . 6 ⊢ (¬ y ∈ x ↔ (y ∈ x ↔ ¬ y = y))
4	3	albii 997	. . . . 5 ⊢ (∀y ¬ y ∈ x ↔ ∀y(y ∈ x ↔ ¬ y = y))
5	4	exbii 1049	. . . 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-17 969	. . . 4 ⊢ (¬ y = y → ∀x ¬ y = y)
8	7	bm1.1 1460	. . 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 1385	. 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 952   = wceq 954   ∈ wcel 956  ∃wex 978  ∃!weu 1378

This theorem is referenced by:  0ex 2706  snex 2745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380

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