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Theorem riotacl2 7366
Description: Membership law for "the unique element in 𝐴 such that 𝜑". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
riotacl2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})

Proof of Theorem riotacl2
StepHypRef Expression
1 df-reu 3376 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotacl 6518 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 7349 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 3432 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
63, 4, 53eltr4g 2849 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  ∃!weu 2561  {cab 2708  ∃!wreu 3373  {crab 3431  cio 6482  crio 7348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-un 3949  df-in 3951  df-ss 3961  df-sn 4623  df-pr 4625  df-uni 4902  df-iota 6484  df-riota 7349
This theorem is referenced by:  riotacl  7367  riotasbc  7368  riotaxfrd  7384  supub  9436  suplub  9437  ordtypelem3  9497  catlid  17609  catrid  17610  grplinv  18849  pj1id  19531  evlsval2  21579  ig1pval3  25621  coelem  25669  quotlem  25742  mircgr  27773  mirbtwn  27774  grpoidinv2  29631  grpoinv  29641  cnlnadjlem5  31187  cvmsiota  34097  cvmliftiota  34121  mpaalem  41663  disjinfi  43660
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