Theorem iotan0aiotaex 44585

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

Step	Hyp	Ref	Expression
1		iotanul 6411	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
2	1	necon1ai 2971	. 2 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3		aiotaexb 44581	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
4	2, 3	sylib 217	1 ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  ∃!weu 2568   ≠ wne 2943  Vcvv 3432  ∅c0 4256  ℩cio 6389  ℩'caiota 44575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-uni 4840  df-int 4880  df-iota 6391  df-aiota 44577

This theorem is referenced by: (None)