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Theorem riotacl2 5405
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 (∃!x A φ → (x A φ) {x Aφ})

Proof of Theorem riotacl2
StepHypRef Expression
1 df-reu 2291 . . 3 (∃!x A φ∃!x(x A φ))
2 iotacl 4817 . . 3 (∃!x(x A φ) → (℩x(x A φ)) {x ∣ (x A φ)})
31, 2sylbi 114 . 2 (∃!x A φ → (℩x(x A φ)) {x ∣ (x A φ)})
4 df-riota 5393 . 2 (x A φ) = (℩x(x A φ))
5 df-rab 2293 . 2 {x Aφ} = {x ∣ (x A φ)}
63, 4, 53eltr4g 2105 1 (∃!x A φ → (x A φ) {x Aφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  ∃!weu 1882  {cab 2008  ∃!wreu 2286  {crab 2288  cio 4792  crio 5392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794  df-riota 5393
This theorem is referenced by:  riotacl  5406  riotasbc  5407
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