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Theorem riotacl2 7384
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 3375 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotacl 6528 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 7367 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 3431 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
63, 4, 53eltr4g 2848 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  ∃!weu 2560  {cab 2707  ∃!wreu 3372  {crab 3430  cio 6492  crio 7366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6494  df-riota 7367
This theorem is referenced by:  riotacl  7385  riotasbc  7386  riotaxfrd  7402  supub  9456  suplub  9457  ordtypelem3  9517  catlid  17631  catrid  17632  grplinv  18910  pj1id  19608  evlsval2  21869  ig1pval3  25927  coelem  25975  quotlem  26049  mircgr  28175  mirbtwn  28176  grpoidinv2  30035  grpoinv  30045  cnlnadjlem5  31591  cvmsiota  34566  cvmliftiota  34590  mpaalem  42196  disjinfi  44189
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