Theorem zfnuleu 4047

Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2122 to strengthen the hypothesis in the form of axnul 4048). (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 1342	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥))
3	2	albii 1446	. . . . 5 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
4	3	exbii 1584	. . . 4 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
5	1, 4	mpbi 144	. . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
6		nfv 1508	. . . 4 ⊢ Ⅎ𝑥⊥
7	6	bm1.1 2122	. . 3 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
8	5, 7	ax-mp 5	. 2 ⊢ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
9	3	eubii 2006	. 2 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
10	8, 9	mpbir 145	1 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1329  ⊥wfal 1336  ∃wex 1468  ∃!weu 1997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000

This theorem is referenced by: (None)