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Theorem iota0ndef 43618
 Description: Example for an undefined iota being the empty set, i.e., ∀𝑦𝑦 ∈ 𝑥 is a wff not satisfied by a (unique) value 𝑥 (there is no set, and therefore certainly no unique set, which contains every set). (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
iota0ndef (℩𝑥𝑦 𝑦𝑥) = ∅
Distinct variable group:   𝑥,𝑦

Proof of Theorem iota0ndef
StepHypRef Expression
1 nalset 5184 . . . 4 ¬ ∃𝑥𝑦 𝑦𝑥
21intnanr 491 . . 3 ¬ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥)
3 df-eu 2632 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥))
42, 3mtbir 326 . 2 ¬ ∃!𝑥𝑦 𝑦𝑥
5 iotanul 6306 . 2 (¬ ∃!𝑥𝑦 𝑦𝑥 → (℩𝑥𝑦 𝑦𝑥) = ∅)
64, 5ax-mp 5 1 (℩𝑥𝑦 𝑦𝑥) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781  ∃*wmo 2599  ∃!weu 2631  ∅c0 4246  ℩cio 6285 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-uni 4804  df-iota 6287 This theorem is referenced by: (None)
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