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Theorem iota0ndef 46989
Description: Example for an undefined iota being the empty set, i.e., 𝑦𝑦𝑥 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 5319 . . . 4 ¬ ∃𝑥𝑦 𝑦𝑥
21intnanr 487 . . 3 ¬ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥)
3 df-eu 2567 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥))
42, 3mtbir 323 . 2 ¬ ∃!𝑥𝑦 𝑦𝑥
5 iotanul 6541 . 2 (¬ ∃!𝑥𝑦 𝑦𝑥 → (℩𝑥𝑦 𝑦𝑥) = ∅)
64, 5ax-mp 5 1 (℩𝑥𝑦 𝑦𝑥) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535   = wceq 1537  wex 1776  ∃*wmo 2536  ∃!weu 2566  c0 4339  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913  df-iota 6516
This theorem is referenced by: (None)
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