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Theorem snriota 5713
Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
Assertion
Ref Expression
snriota (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})

Proof of Theorem snriota
StepHypRef Expression
1 df-reu 2397 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 sniota 5073 . . 3 (∃!𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
31, 2sylbi 120 . 2 (∃!𝑥𝐴 𝜑 → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
4 df-rab 2399 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-riota 5684 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
65sneqi 3505 . 2 {(𝑥𝐴 𝜑)} = {(℩𝑥(𝑥𝐴𝜑))}
73, 4, 63eqtr4g 2172 1 (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  ∃!weu 1975  {cab 2101  ∃!wreu 2392  {crab 2394  {csn 3493  cio 5044  crio 5683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-sn 3499  df-pr 3500  df-uni 3703  df-iota 5046  df-riota 5684
This theorem is referenced by:  divalgmod  11472
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