Theorem iotan0aiotaex 47830

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 6516	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
2	1	necon1ai 2985	. 2 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3		aiotaexb 47826	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
4	2, 3	sylib 221	1 ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  ∃!weu 2596   ≠ wne 2958  Vcvv 3455  ∅c0 4286  ℩cio 6490  ℩'caiota 47820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-nul 4287  df-sn 4590  df-uni 4873  df-int 4913  df-iota 6492  df-aiota 47822

This theorem is referenced by: (None)