Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-nuliota Structured version   Visualization version   GIF version

Theorem bj-nuliota 35926
Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 35927. (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 2579 . . . . 5 (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥𝑦 ¬ 𝑦𝑥)
52, 4mpbi 229 . . . 4 ∃!𝑥𝑦 ¬ 𝑦𝑥
6 eleq2 2822 . . . . . . 7 (𝑥 = ∅ → (𝑦𝑥𝑦 ∈ ∅))
76notbid 317 . . . . . 6 (𝑥 = ∅ → (¬ 𝑦𝑥 ↔ ¬ 𝑦 ∈ ∅))
87albidv 1923 . . . . 5 (𝑥 = ∅ → (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
98iota2 6529 . . . 4 ((∅ ∈ V ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅))
101, 5, 9mp2an 690 . . 3 (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅)
11 noel 4329 . . 3 ¬ 𝑦 ∈ ∅
1210, 11mpgbi 1800 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅
1312eqcomi 2741 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1539   = wceq 1541  wcel 2106  ∃!weu 2562  Vcvv 3474  c0 4321  cio 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  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 6492
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator