Theorem riotaund 6838

Description: Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)

Assertion

Ref	Expression
riotaund	⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaund

Step	Hyp	Ref	Expression
1		df-riota 6802	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-reu 3061	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		iotanul 6045	. . 3 ⊢ (¬ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∅)
4	2, 3	sylnbi 321	. 2 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∅)
5	1, 4	syl5eq 2810	1 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  ∃!weu 2580  ∃!wreu 3056  ∅c0 4078  ℩cio 6028  ℩crio 6801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-reu 3061  df-v 3351  df-dif 3734  df-in 3738  df-ss 3745  df-nul 4079  df-sn 4334  df-uni 4594  df-iota 6030  df-riota 6802

This theorem is referenced by:  riotassuni  6839  riotaclb  6840  supval2  8567  lubval  17251  glbval  17264  finxpreclem4  33596