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Theorem bj-nuliota 34755
 Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 34756. (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 5178 . . . 4 ∅ ∈ V
21eueqi 3624 . . . . 5 ∃!𝑥 𝑥 = ∅
3 eq0 4243 . . . . . 6 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43eubii 2605 . . . . 5 (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥𝑦 ¬ 𝑦𝑥)
52, 4mpbi 233 . . . 4 ∃!𝑥𝑦 ¬ 𝑦𝑥
6 eleq2 2841 . . . . . . 7 (𝑥 = ∅ → (𝑦𝑥𝑦 ∈ ∅))
76notbid 322 . . . . . 6 (𝑥 = ∅ → (¬ 𝑦𝑥 ↔ ¬ 𝑦 ∈ ∅))
87albidv 1922 . . . . 5 (𝑥 = ∅ → (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
98iota2 6325 . . . 4 ((∅ ∈ V ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅))
101, 5, 9mp2an 692 . . 3 (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅)
11 noel 4231 . . 3 ¬ 𝑦 ∈ ∅
1210, 11mpgbi 1801 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅
1312eqcomi 2768 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1537   = wceq 1539   ∈ wcel 2112  ∃!weu 2588  Vcvv 3410  ∅c0 4226  ℩cio 6293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-sn 4524  df-pr 4526  df-uni 4800  df-iota 6295 This theorem is referenced by: (None)
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