Theorem bj-nuliota 33969

Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 33970. (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 5109	. . . 4 ⊢ ∅ ∈ V
2	1	eueqi 3641	. . . . 5 ⊢ ∃!𝑥 𝑥 = ∅
3		eq0 4234	. . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	eubii 2632	. . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
5	2, 4	mpbi 231	. . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
6		eleq2 2873	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅))
7	6	notbid 319	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅))
8	7	albidv 1902	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
9	8	iota2 6222	. . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅))
10	1, 5, 9	mp2an 688	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)
11		noel 4222	. . 3 ⊢ ¬ 𝑦 ∈ ∅
12	10, 11	mpgbi 1784	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅
13	12	eqcomi 2806	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 207  ∀wal 1523   = wceq 1525   ∈ wcel 2083  ∃!weu 2613  Vcvv 3440  ∅c0 4217  ℩cio 6194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-nul 5108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-nul 4218  df-sn 4479  df-pr 4481  df-uni 4752  df-iota 6196

This theorem is referenced by: (None)