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Theorem snriota 7416
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 3375 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 sniota 6544 . . 3 (∃!𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
4 df-rab 3431 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-riota 7382 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
65sneqi 4643 . 2 {(𝑥𝐴 𝜑)} = {(℩𝑥(𝑥𝐴𝜑))}
73, 4, 63eqtr4g 2793 1 (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  ∃!weu 2557  {cab 2705  ∃!wreu 3372  {crab 3430  {csn 4632  cio 6503  crio 7381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-un 3954  df-in 3956  df-ss 3966  df-sn 4633  df-pr 4635  df-uni 4913  df-iota 6505  df-riota 7382
This theorem is referenced by:  divalgmod  16390
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