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Theorem snriota 7421
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 3379 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 sniota 6554 . . 3 (∃!𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
31, 2sylbi 217 . 2 (∃!𝑥𝐴 𝜑 → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
4 df-rab 3434 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-riota 7388 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
65sneqi 4642 . 2 {(𝑥𝐴 𝜑)} = {(℩𝑥(𝑥𝐴𝜑))}
73, 4, 63eqtr4g 2800 1 (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ∃!weu 2566  {cab 2712  ∃!wreu 3376  {crab 3433  {csn 4631  cio 6514  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  divalgmod  16440
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