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Theorem iotan0aiotaex 47042
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 6540 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
21necon1ai 2965 . 2 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3 aiotaexb 47038 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
42, 3sylib 218 1 ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  ∃!weu 2565  wne 2937  Vcvv 3477  c0 4338  cio 6513  ℩'caiota 47032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-in 3969  df-ss 3979  df-nul 4339  df-sn 4631  df-uni 4912  df-int 4951  df-iota 6515  df-aiota 47034
This theorem is referenced by: (None)
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