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Theorem riotacl2 7310
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 3350 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotacl 6465 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 7293 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 3404 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
63, 4, 53eltr4g 2854 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  ∃!weu 2566  {cab 2713  ∃!wreu 3347  {crab 3403  cio 6429  crio 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-un 3903  df-in 3905  df-ss 3915  df-sn 4574  df-pr 4576  df-uni 4853  df-iota 6431  df-riota 7293
This theorem is referenced by:  riotacl  7311  riotasbc  7312  riotaxfrd  7328  supub  9316  suplub  9317  ordtypelem3  9377  catlid  17489  catrid  17490  grplinv  18724  pj1id  19400  evlsval2  21403  ig1pval3  25445  coelem  25493  quotlem  25566  mircgr  27307  mirbtwn  27308  grpoidinv2  29165  grpoinv  29175  cnlnadjlem5  30721  cvmsiota  33538  cvmliftiota  33562  mpaalem  41240  disjinfi  43058
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