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Theorem sniota 5876
Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
sniota (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})

Proof of Theorem sniota
StepHypRef Expression
1 nfeu1 2479 . 2 𝑥∃!𝑥𝜑
2 nfab1 2765 . 2 𝑥{𝑥𝜑}
3 nfiota1 5851 . . 3 𝑥(℩𝑥𝜑)
43nfsn 4240 . 2 𝑥{(℩𝑥𝜑)}
5 iota1 5863 . . . 4 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6 eqcom 2628 . . . 4 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
75, 6syl6bb 276 . . 3 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
8 abid 2609 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 velsn 4191 . . 3 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
107, 8, 93bitr4g 303 . 2 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
111, 2, 4, 10eqrd 3620 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  ∃!weu 2469  {cab 2607  {csn 4175  cio 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-sbc 3434  df-un 3577  df-sn 4176  df-pr 4178  df-uni 4435  df-iota 5849
This theorem is referenced by:  snriota  6638
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