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Theorem riotaund 6632
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 6596 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
2 df-reu 2916 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iotanul 5854 . . 3 (¬ ∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) = ∅)
42, 3sylnbi 320 . 2 (¬ ∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) = ∅)
51, 4syl5eq 2666 1 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  ∃!weu 2468  ∃!wreu 2911  c0 3907  cio 5837  crio 6595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-reu 2916  df-v 3197  df-dif 3570  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-uni 4428  df-iota 5839  df-riota 6596
This theorem is referenced by:  riotassuni  6633  riotaclb  6634  supval2  8346  lubval  16965  glbval  16978  finxpreclem4  33202
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