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Theorem sniota 4821
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 1893 . . 3 x∃!xφ
2 iota1 4808 . . . . 5 (∃!xφ → (φ ↔ (℩xφ) = x))
3 eqcom 2024 . . . . 5 ((℩xφ) = xx = (℩xφ))
42, 3syl6bb 185 . . . 4 (∃!xφ → (φx = (℩xφ)))
5 abid 2010 . . . 4 (x {xφ} ↔ φ)
6 vex 2538 . . . . 5 x V
76elsnc 3373 . . . 4 (x {(℩xφ)} ↔ x = (℩xφ))
84, 5, 73bitr4g 212 . . 3 (∃!xφ → (x {xφ} ↔ x {(℩xφ)}))
91, 8alrimi 1396 . 2 (∃!xφx(x {xφ} ↔ x {(℩xφ)}))
10 nfab1 2162 . . 3 x{xφ}
11 nfiota1 4796 . . . 4 x(℩xφ)
1211nfsn 3404 . . 3 x{(℩xφ)}
1310, 12cleqf 2183 . 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 1226   = wceq 1228   wcel 1374  ∃!weu 1882  {cab 2008  {csn 3350  cio 4792
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794
This theorem is referenced by:  snriota  5421
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