Theorem iotan0aiotaex 43284

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 6328	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
2	1	necon1ai 3043	. 2 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3		aiotaexb 43282	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
4	2, 3	sylib 220	1 ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2110  ∃!weu 2649   ≠ wne 3016  Vcvv 3495  ∅c0 4291  ℩cio 6307  ℩'caiota 43276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4562  df-uni 4833  df-int 4870  df-iota 6309  df-aiota 43278

This theorem is referenced by: (None)