Theorem aiotaexb 48267

Description: The alternate iota over a wff 𝜑 is a set iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
aiotaexb	⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Proof of Theorem aiotaexb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intexab 5307	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
2		euabsn2 4682	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3		df-aiota 48263	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	3	eleq1i 2851	. 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	1, 2, 4	3bitr4i 306	1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃!weu 2593  {cab 2738  Vcvv 3450  {csn 4584  ∩ cint 4906  ℩'caiota 48261

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-nul 4278  df-sn 4585  df-int 4907  df-aiota 48263

This theorem is used by:  aiotavb  48268  iotan0aiotaex  48271  aiotaexaiotaiota  48272  aiota0ndef  48275