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 33317
Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 33318. (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 4934 . . . 4 ∅ ∈ V
21eueq1 3512 . . . . 5 ∃!𝑥 𝑥 = ∅
3 eq0 4064 . . . . . 6 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43eubii 2621 . . . . 5 (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥𝑦 ¬ 𝑦𝑥)
52, 4mpbi 220 . . . 4 ∃!𝑥𝑦 ¬ 𝑦𝑥
6 eleq2 2820 . . . . . . 7 (𝑥 = ∅ → (𝑦𝑥𝑦 ∈ ∅))
76notbid 307 . . . . . 6 (𝑥 = ∅ → (¬ 𝑦𝑥 ↔ ¬ 𝑦 ∈ ∅))
87albidv 1990 . . . . 5 (𝑥 = ∅ → (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
98iota2 6030 . . . 4 ((∅ ∈ V ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅))
101, 5, 9mp2an 710 . . 3 (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅)
11 noel 4054 . . 3 ¬ 𝑦 ∈ ∅
1210, 11mpgbi 1866 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅
1312eqcomi 2761 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1622   = wceq 1624  wcel 2131  ∃!weu 2599  Vcvv 3332  c0 4050  cio 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-nul 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-nul 4051  df-sn 4314  df-pr 4316  df-uni 4581  df-iota 6004
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator