MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sniota Structured version   Visualization version   GIF version

Theorem sniota 6221
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 2635 . 2 𝑥∃!𝑥𝜑
2 nfab1 2951 . 2 𝑥{𝑥𝜑}
3 nfiota1 6196 . . 3 𝑥(℩𝑥𝜑)
43nfsn 4554 . 2 𝑥{(℩𝑥𝜑)}
5 iota1 6208 . . . 4 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6 eqcom 2802 . . . 4 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
75, 6syl6bb 288 . . 3 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
8 abid 2779 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
9 velsn 4492 . . 3 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
107, 8, 93bitr4g 315 . 2 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
111, 2, 4, 10eqrd 3912 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  wcel 2081  ∃!weu 2611  {cab 2775  {csn 4476  cio 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439  df-sbc 3710  df-un 3868  df-sn 4477  df-pr 4479  df-uni 4750  df-iota 6194
This theorem is referenced by:  snriota  7012  fnimasnd  38676
  Copyright terms: Public domain W3C validator