Theorem iota0ndef 47743

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 5276	. . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2	1	intnanr 492	. . 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		iotanul 6516	. 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 400  ∀wal 1566   = wceq 1568  ∃wex 1807  ∃*wmo 2563  ∃!weu 2594  ∅c0 4285  ℩cio 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-ss 3921  df-nul 4286  df-sn 4589  df-uni 4872  df-iota 6492

This theorem is referenced by: (None)