Theorem bj-nuliota 37637

Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 37638. (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 5269	. . . 4 ⊢ ∅ ∈ V
2	1	eueqi 3671	. . . . 5 ⊢ ∃!𝑥 𝑥 = ∅
3		eq0 4303	. . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	eubii 2611	. . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
5	2, 4	mpbi 233	. . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
6		eleq2 2850	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅))
7	6	notbid 321	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅))
8	7	albidv 1948	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
9	8	iota2 6525	. . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅))
10	1, 5, 9	mp2an 704	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)
11		noel 4290	. . 3 ⊢ ¬ 𝑦 ∈ ∅
12	10, 11	mpgbi 1826	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅
13	12	eqcomi 2770	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1566   = wceq 1568   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453  ∅c0 4285  ℩cio 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4589  df-pr 4591  df-uni 4872  df-iota 6492

This theorem is referenced by: (None)