Theorem aiota0ndef 47102

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 47044, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
aiota0ndef	⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V

Distinct variable group:   𝑥,𝑦

Proof of Theorem aiota0ndef

Step	Hyp	Ref	Expression
1		nalset 5271	. . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2	1	intnanr 487	. . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)
3		df-eu 2563	. . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥))
4	2, 3	mtbir 323	. 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥
5		df-nel 3031	. . 3 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V)
6		aiotaexb 47094	. . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V)
7	5, 6	xchbinxr 335	. 2 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥)
8	4, 7	mpbir 231	1 ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1538  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2532  ∃!weu 2562   ∉ wnel 3030  Vcvv 3450  ℩'caiota 47088

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300  df-sn 4593  df-int 4914  df-aiota 47090

This theorem is referenced by: (None)