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Theorem aiota0ndef 46105
Description: Example for an undefined alternate iota being no 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). This is different from iota0ndef 46049, 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 5314 . . . 4 ¬ ∃𝑥𝑦 𝑦𝑥
21intnanr 487 . . 3 ¬ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥)
3 df-eu 2562 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥))
42, 3mtbir 322 . 2 ¬ ∃!𝑥𝑦 𝑦𝑥
5 df-nel 3046 . . 3 ((℩'𝑥𝑦 𝑦𝑥) ∉ V ↔ ¬ (℩'𝑥𝑦 𝑦𝑥) ∈ V)
6 aiotaexb 46097 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (℩'𝑥𝑦 𝑦𝑥) ∈ V)
75, 6xchbinxr 334 . 2 ((℩'𝑥𝑦 𝑦𝑥) ∉ V ↔ ¬ ∃!𝑥𝑦 𝑦𝑥)
84, 7mpbir 230 1 (℩'𝑥𝑦 𝑦𝑥) ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538  wex 1780  wcel 2105  ∃*wmo 2531  ∃!weu 2561  wnel 3045  Vcvv 3473  ℩'caiota 46091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-int 4952  df-aiota 46093
This theorem is referenced by: (None)
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