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Theorem iotan0aiotaex 46532
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 6521 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
21necon1ai 2958 . 2 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3 aiotaexb 46528 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
42, 3sylib 217 1 ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  ∃!weu 2556  wne 2930  Vcvv 3463  c0 4319  cio 6493  ℩'caiota 46522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4320  df-sn 4626  df-uni 4905  df-int 4946  df-iota 6495  df-aiota 46524
This theorem is referenced by: (None)
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