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Theorem bj-nuliota 36241
Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 36242. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nuliota ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nuliota
StepHypRef Expression
1 0ex 5306 . . . 4 ∅ ∈ V
21eueqi 3704 . . . . 5 ∃!𝑥 𝑥 = ∅
3 eq0 4342 . . . . . 6 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43eubii 2577 . . . . 5 (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥𝑦 ¬ 𝑦𝑥)
52, 4mpbi 229 . . . 4 ∃!𝑥𝑦 ¬ 𝑦𝑥
6 eleq2 2820 . . . . . . 7 (𝑥 = ∅ → (𝑦𝑥𝑦 ∈ ∅))
76notbid 317 . . . . . 6 (𝑥 = ∅ → (¬ 𝑦𝑥 ↔ ¬ 𝑦 ∈ ∅))
87albidv 1921 . . . . 5 (𝑥 = ∅ → (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
98iota2 6531 . . . 4 ((∅ ∈ V ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅))
101, 5, 9mp2an 688 . . 3 (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅)
11 noel 4329 . . 3 ¬ 𝑦 ∈ ∅
1210, 11mpgbi 1798 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅
1312eqcomi 2739 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1537   = wceq 1539  wcel 2104  ∃!weu 2560  Vcvv 3472  c0 4321  cio 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6494
This theorem is referenced by: (None)
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