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Theorem sniota 4837
Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
sniota (∃!xφ → {xφ} = {(℩xφ)})

Proof of Theorem sniota
StepHypRef Expression
1 nfeu1 1908 . . 3 x∃!xφ
2 iota1 4824 . . . . 5 (∃!xφ → (φ ↔ (℩xφ) = x))
3 eqcom 2039 . . . . 5 ((℩xφ) = xx = (℩xφ))
42, 3syl6bb 185 . . . 4 (∃!xφ → (φx = (℩xφ)))
5 abid 2025 . . . 4 (x {xφ} ↔ φ)
6 vex 2554 . . . . 5 x V
76elsnc 3390 . . . 4 (x {(℩xφ)} ↔ x = (℩xφ))
84, 5, 73bitr4g 212 . . 3 (∃!xφ → (x {xφ} ↔ x {(℩xφ)}))
91, 8alrimi 1412 . 2 (∃!xφx(x {xφ} ↔ x {(℩xφ)}))
10 nfab1 2177 . . 3 x{xφ}
11 nfiota1 4812 . . . 4 x(℩xφ)
1211nfsn 3421 . . 3 x{(℩xφ)}
1310, 12cleqf 2198 . 2 ({xφ} = {(℩xφ)} ↔ x(x {xφ} ↔ x {(℩xφ)}))
149, 13sylibr 137 1 (∃!xφ → {xφ} = {(℩xφ)})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  ∃!weu 1897  {cab 2023  {csn 3367  cio 4808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  snriota  5440
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