Theorem iota0ndef 44573

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 5240	. . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2	1	intnanr 487	. . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)
3		df-eu 2564	. . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥))
4	2, 3	mtbir 322	. 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥
5		iotanul 6425	. 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 1535   = wceq 1537  ∃wex 1777  ∃*wmo 2533  ∃!weu 2563  ∅c0 4259  ℩cio 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-v 3436  df-dif 3892  df-in 3896  df-ss 3906  df-nul 4260  df-sn 4565  df-uni 4842  df-iota 6399

This theorem is referenced by: (None)