Theorem zfnuleu 4157

Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2181 to strengthen the hypothesis in the form of axnul 4158). (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 1375	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥))
3	2	albii 1484	. . . . 5 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
4	3	exbii 1619	. . . 4 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
5	1, 4	mpbi 145	. . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
6		nfv 1542	. . . 4 ⊢ Ⅎ𝑥⊥
7	6	bm1.1 2181	. . 3 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
8	5, 7	ax-mp 5	. 2 ⊢ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
9	3	eubii 2054	. 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 1362  ⊥wfal 1369  ∃wex 1506  ∃!weu 2045

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by: (None)