Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  snriota GIF version

Theorem snriota 5497
 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 2313 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 sniota 4894 . . 3 (∃!𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
31, 2sylbi 114 . 2 (∃!𝑥𝐴 𝜑 → {𝑥 ∣ (𝑥𝐴𝜑)} = {(℩𝑥(𝑥𝐴𝜑))})
4 df-rab 2315 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
5 df-riota 5468 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
65sneqi 3387 . 2 {(𝑥𝐴 𝜑)} = {(℩𝑥(𝑥𝐴𝜑))}
73, 4, 63eqtr4g 2097 1 (∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∃!weu 1900  {cab 2026  ∃!wreu 2308  {crab 2310  {csn 3375  ℩cio 4865  ℩crio 5467 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867  df-riota 5468 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator