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Theorem riotaund 6687
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 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaund
StepHypRef Expression
1 df-riota 6651 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
2 df-reu 2948 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iotanul 5904 . . 3 (¬ ∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) = ∅)
42, 3sylnbi 319 . 2 (¬ ∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) = ∅)
51, 4syl5eq 2697 1 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  ∃!weu 2498  ∃!wreu 2943  c0 3948  cio 5887  crio 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469  df-iota 5889  df-riota 6651
This theorem is referenced by:  riotassuni  6688  riotaclb  6689  supval2  8402  lubval  17031  glbval  17044  finxpreclem4  33361
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