Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aiota0ndef Structured version   Visualization version   GIF version

Theorem aiota0ndef 43578
 Description: Example for an undefined alternate iota being no 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). This is different from iota0ndef 43557, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiota0ndef (℩'𝑥𝑦 𝑦𝑥) ∉ V
Distinct variable group:   𝑥,𝑦

Proof of Theorem aiota0ndef
StepHypRef Expression
1 nalset 5203 . . . 4 ¬ ∃𝑥𝑦 𝑦𝑥
21intnanr 491 . . 3 ¬ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥)
3 df-eu 2655 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥))
42, 3mtbir 326 . 2 ¬ ∃!𝑥𝑦 𝑦𝑥
5 df-nel 3119 . . 3 ((℩'𝑥𝑦 𝑦𝑥) ∉ V ↔ ¬ (℩'𝑥𝑦 𝑦𝑥) ∈ V)
6 aiotaexb 43572 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (℩'𝑥𝑦 𝑦𝑥) ∈ V)
75, 6xchbinxr 338 . 2 ((℩'𝑥𝑦 𝑦𝑥) ∉ V ↔ ¬ ∃!𝑥𝑦 𝑦𝑥)
84, 7mpbir 234 1 (℩'𝑥𝑦 𝑦𝑥) ∉ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2115  ∃*wmo 2622  ∃!weu 2654   ∉ wnel 3118  Vcvv 3480  ℩'caiota 43566 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-int 4863  df-aiota 43568 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator