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Theorem riotacl2 5603
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 2366 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotacl 4990 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 119 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 5590 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 2368 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
63, 4, 53eltr4g 2173 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  ∃!weu 1948  {cab 2074  ∃!wreu 2361  {crab 2363  cio 4965  crio 5589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967  df-riota 5590
This theorem is referenced by:  riotacl  5604  riotasbc  5605  supubti  6673  suplubti  6674
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