Theorem iota0ndef 47499

Description: Example for an undefined iota being the empty 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). (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
iota0ndef	⊢ (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅

Distinct variable group:   𝑥,𝑦

Proof of Theorem iota0ndef

Step	Hyp	Ref	Expression
1		nalset 5239	. . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2	1	intnanr 488	. . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)
3		df-eu 2569	. . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥))
4	2, 3	mtbir 324	. 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥
5		iotanul 6468	. 2 ⊢ (¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥 → (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅)
6	4, 5	ax-mp 5	1 ⊢ (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 396  ∀wal 1541   = wceq 1543  ∃wex 1782  ∃*wmo 2537  ∃!weu 2568  ∅c0 4264  ℩cio 6442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3430  df-dif 3889  df-ss 3903  df-nul 4265  df-sn 4559  df-uni 4842  df-iota 6444

This theorem is referenced by: (None)