Theorem zfnuleu 4129

Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2162 to strengthen the hypothesis in the form of axnul 4130). (Contributed by NM, 22-Dec-2007.)

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		nbfal 1364	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥))
3	2	albii 1470	. . . . 5 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
4	3	exbii 1605	. . . 4 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
5	1, 4	mpbi 145	. . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
6		nfv 1528	. . . 4 ⊢ Ⅎ𝑥⊥
7	6	bm1.1 2162	. . 3 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
8	5, 7	ax-mp 5	. 2 ⊢ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
9	3	eubii 2035	. 2 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
10	8, 9	mpbir 146	1 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1351  ⊥wfal 1358  ∃wex 1492  ∃!weu 2026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029

This theorem is referenced by: (None)