Theorem aiota0ndef 48111

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 48053, 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 5268	. . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2	1	intnanr 493	. . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)
3		df-eu 2595	. . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥))
4	2, 3	mtbir 326	. 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥
5		df-nel 3063	. . 3 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V)
6		aiotaexb 48103	. . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V)
7	5, 6	xchbinxr 338	. 2 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥)
8	4, 7	mpbir 234	1 ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  ∃!weu 2594   ∉ wnel 3062  Vcvv 3451  ℩'caiota 48097

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280  df-sn 4585  df-int 4908  df-aiota 48099

This theorem is used by: (None)