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Theorem iotan0aiotaex 46099
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 6520 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
21necon1ai 2966 . 2 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3 aiotaexb 46095 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
42, 3sylib 217 1 ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  ∃!weu 2560  wne 2938  Vcvv 3472  c0 4321  cio 6492  ℩'caiota 46089
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-uni 4908  df-int 4950  df-iota 6494  df-aiota 46091
This theorem is referenced by: (None)
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