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Theorem aiota0ndef 47109
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 47051, 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 5313 . . . 4 ¬ ∃𝑥𝑦 𝑦𝑥
21intnanr 487 . . 3 ¬ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥)
3 df-eu 2569 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (∃𝑥𝑦 𝑦𝑥 ∧ ∃*𝑥𝑦 𝑦𝑥))
42, 3mtbir 323 . 2 ¬ ∃!𝑥𝑦 𝑦𝑥
5 df-nel 3047 . . 3 ((℩'𝑥𝑦 𝑦𝑥) ∉ V ↔ ¬ (℩'𝑥𝑦 𝑦𝑥) ∈ V)
6 aiotaexb 47101 . . 3 (∃!𝑥𝑦 𝑦𝑥 ↔ (℩'𝑥𝑦 𝑦𝑥) ∈ V)
75, 6xchbinxr 335 . 2 ((℩'𝑥𝑦 𝑦𝑥) ∉ V ↔ ¬ ∃!𝑥𝑦 𝑦𝑥)
84, 7mpbir 231 1 (℩'𝑥𝑦 𝑦𝑥) ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  wnel 3046  Vcvv 3480  ℩'caiota 47095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-int 4947  df-aiota 47097
This theorem is referenced by: (None)
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