Theorem iota0ndef 47035

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 5288	. . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2	1	intnanr 487	. . 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 323	. 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥
5		iotanul 6514	. 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 395  ∀wal 1538   = wceq 1540  ∃wex 1779  ∃*wmo 2538  ∃!weu 2568  ∅c0 4313  ℩cio 6487

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 2708  ax-sep 5271

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-v 3466  df-dif 3934  df-ss 3948  df-nul 4314  df-sn 4607  df-uni 4889  df-iota 6489

This theorem is referenced by: (None)