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Theorem sniota 6554
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 2586 . 2 𝑥∃!𝑥𝜑
2 nfab1 2905 . 2 𝑥{𝑥𝜑}
3 nfiota1 6518 . . 3 𝑥(℩𝑥𝜑)
43nfsn 4712 . 2 𝑥{(℩𝑥𝜑)}
5 iota1 6540 . . . 4 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6 eqcom 2742 . . . 4 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
75, 6bitrdi 287 . . 3 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
8 abid 2716 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 velsn 4647 . . 3 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
107, 8, 93bitr4g 314 . 2 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
111, 2, 4, 10eqrd 4015 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  ∃!weu 2566  {cab 2712  {csn 4631  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by:  snriota  7421
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