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Theorem bj-nuliota 32658
Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 32659. (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 4755 . . . 4 ∅ ∈ V
21eueq1 3366 . . . . 5 ∃!𝑥 𝑥 = ∅
3 eq0 3910 . . . . . 6 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43eubii 2496 . . . . 5 (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥𝑦 ¬ 𝑦𝑥)
52, 4mpbi 220 . . . 4 ∃!𝑥𝑦 ¬ 𝑦𝑥
6 eleq2 2693 . . . . . . 7 (𝑥 = ∅ → (𝑦𝑥𝑦 ∈ ∅))
76notbid 308 . . . . . 6 (𝑥 = ∅ → (¬ 𝑦𝑥 ↔ ¬ 𝑦 ∈ ∅))
87albidv 1851 . . . . 5 (𝑥 = ∅ → (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅))
98iota2 5839 . . . 4 ((∅ ∈ V ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅))
101, 5, 9mp2an 707 . . 3 (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅)
11 noel 3900 . . 3 ¬ 𝑦 ∈ ∅
1210, 11mpgbi 1722 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) = ∅
1312eqcomi 2635 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1478   = wceq 1480  wcel 1992  ∃!weu 2474  Vcvv 3191  c0 3896  cio 5811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5813
This theorem is referenced by: (None)
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