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Theorem iotan0aiotaex 47105
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 6539 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
21necon1ai 2968 . 2 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3 aiotaexb 47101 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
42, 3sylib 218 1 ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  ∃!weu 2568  wne 2940  Vcvv 3480  c0 4333  cio 6512  ℩'caiota 47095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-uni 4908  df-int 4947  df-iota 6514  df-aiota 47097
This theorem is referenced by: (None)
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