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Theorem iotan0aiotaex 43816
 Description: If the iota over a wff 𝜑 is not empty, the alternate iota over 𝜑 is a set. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
iotan0aiotaex ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)

Proof of Theorem iotan0aiotaex
StepHypRef Expression
1 iotanul 6310 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
21necon1ai 3014 . 2 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3 aiotaexb 43814 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
42, 3sylib 221 1 ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ∃!weu 2628   ≠ wne 2987  Vcvv 3442  ∅c0 4246  ℩cio 6289  ℩'caiota 43808 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4247  df-sn 4529  df-uni 4805  df-int 4843  df-iota 6291  df-aiota 43810 This theorem is referenced by: (None)
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