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Theorem riotacl2 6578
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 2914 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotacl 5833 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 207 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 6565 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 2916 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
63, 4, 53eltr4g 2715 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  ∃!weu 2469  {cab 2607  ∃!wreu 2909  {crab 2911  cio 5808  crio 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-un 3560  df-sn 4149  df-pr 4151  df-uni 4403  df-iota 5810  df-riota 6565
This theorem is referenced by:  riotacl  6579  riotasbc  6580  riotaxfrd  6596  supub  8309  suplub  8310  ordtypelem3  8369  catlid  16265  catrid  16266  grplinv  17389  pj1id  18033  evlsval2  19439  ig1pval3  23838  coelem  23886  quotlem  23959  mircgr  25452  mirbtwn  25453  grpoidinv2  27218  grpoinv  27228  cnlnadjlem5  28779  cvmsiota  30967  cvmliftiota  30991  mpaalem  37203  disjinfi  38854
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