Theorem bj-nuliota 32658

Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 32659. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nuliota	⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nuliota

Step	Hyp	Ref	Expression
1		0ex 4755	. . . 4 ⊢ ∅ ∈ V
2	1	eueq1 3366	. . . . 5 ⊢ ∃!𝑥 𝑥 = ∅
3		eq0 3910	. . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	eubii 2496	. . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
5	2, 4	mpbi 220	. . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
6		eleq2 2693	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅))
7	6	notbid 308	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅))
8	7	albidv 1851	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
9	8	iota2 5839	. . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅))
10	1, 5, 9	mp2an 707	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)
11		noel 3900	. . 3 ⊢ ¬ 𝑦 ∈ ∅
12	10, 11	mpgbi 1722	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅
13	12	eqcomi 2635	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1478   = wceq 1480   ∈ wcel 1992  ∃!weu 2474  Vcvv 3191  ∅c0 3896  ℩cio 5811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5813

This theorem is referenced by: (None)