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Theorem sniota 6091
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 2642 . 2 𝑥∃!𝑥𝜑
2 nfab1 2950 . 2 𝑥{𝑥𝜑}
3 nfiota1 6066 . . 3 𝑥(℩𝑥𝜑)
43nfsn 4434 . 2 𝑥{(℩𝑥𝜑)}
5 iota1 6078 . . . 4 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6 eqcom 2813 . . . 4 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
75, 6syl6bb 278 . . 3 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
8 abid 2794 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 velsn 4386 . . 3 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
107, 8, 93bitr4g 305 . 2 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
111, 2, 4, 10eqrd 3817 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  ∃!weu 2630  {cab 2792  {csn 4370  cio 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-v 3393  df-sbc 3634  df-un 3774  df-sn 4371  df-pr 4373  df-uni 4631  df-iota 6064
This theorem is referenced by:  snriota  6865
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