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Theorem sniota 6551
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 2587 . 2 𝑥∃!𝑥𝜑
2 nfab1 2906 . 2 𝑥{𝑥𝜑}
3 nfiota1 6515 . . 3 𝑥(℩𝑥𝜑)
43nfsn 4706 . 2 𝑥{(℩𝑥𝜑)}
5 iota1 6537 . . . 4 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6 eqcom 2743 . . . 4 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
75, 6bitrdi 287 . . 3 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
8 abid 2717 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 velsn 4641 . . 3 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
107, 8, 93bitr4g 314 . 2 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
111, 2, 4, 10eqrd 4002 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  ∃!weu 2567  {cab 2713  {csn 4625  cio 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-un 3955  df-ss 3967  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513
This theorem is referenced by:  snriota  7422
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