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Theorem riotaund 7135
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 7096 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
2 df-reu 3139 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iotanul 6314 . . 3 (¬ ∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) = ∅)
42, 3sylnbi 333 . 2 (¬ ∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) = ∅)
51, 4syl5eq 2871 1 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  ∃!weu 2654  ∃!wreu 3134  c0 4274  cio 6293  crio 7095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-reu 3139  df-v 3481  df-dif 3921  df-in 3925  df-ss 3935  df-nul 4275  df-sn 4549  df-uni 4820  df-iota 6295  df-riota 7096
This theorem is referenced by:  riotassuni  7136  riotaclb  7137  supval2  8903  lubval  17583  glbval  17596  grpinvfval  18131  finxpreclem4  34713
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