Theorem iotan0aiotaex 48132

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 6517	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
2	1	necon1ai 2983	. 2 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
3		aiotaexb 48128	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
4	2, 3	sylib 221	1 ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∃!weu 2594   ≠ wne 2956  Vcvv 3451  ∅c0 4279  ℩cio 6491  ℩'caiota 48122

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280  df-sn 4585  df-uni 4868  df-int 4908  df-iota 6493  df-aiota 48124

This theorem is used by: (None)