Theorem iotan0aiotaex 45801

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 6522	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
2	1	necon1ai 2969	. 2 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3		aiotaexb 45797	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
4	2, 3	sylib 217	1 ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  ∃!weu 2563   ≠ wne 2941  Vcvv 3475  ∅c0 4323  ℩cio 6494  ℩'caiota 45791

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-uni 4910  df-int 4952  df-iota 6496  df-aiota 45793

This theorem is referenced by: (None)