Theorem bj-nuliota 33019

Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 33020. (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 4790	. . . 4 ⊢ ∅ ∈ V
2	1	eueq1 3379	. . . . 5 ⊢ ∃!𝑥 𝑥 = ∅
3		eq0 3929	. . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	eubii 2492	. . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
5	2, 4	mpbi 220	. . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
6		eleq2 2690	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅))
7	6	notbid 308	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅))
8	7	albidv 1849	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
9	8	iota2 5877	. . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅))
10	1, 5, 9	mp2an 708	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)
11		noel 3919	. . 3 ⊢ ¬ 𝑦 ∈ ∅
12	10, 11	mpgbi 1725	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅
13	12	eqcomi 2631	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  Vcvv 3200  ∅c0 3915  ℩cio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)