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Theorem sniota 4369
 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 → {x φ} = {(℩xφ)})

Proof of Theorem sniota
StepHypRef Expression
1 nfeu1 2214 . . 3 x∃!xφ
2 iota1 4353 . . . . 5 (∃!xφ → (φ ↔ (℩xφ) = x))
3 eqcom 2355 . . . . 5 ((℩xφ) = xx = (℩xφ))
42, 3syl6bb 252 . . . 4 (∃!xφ → (φx = (℩xφ)))
5 abid 2341 . . . 4 (x {x φ} ↔ φ)
6 vex 2862 . . . . 5 x V
76elsnc 3756 . . . 4 (x {(℩xφ)} ↔ x = (℩xφ))
84, 5, 73bitr4g 279 . . 3 (∃!xφ → (x {x φ} ↔ x {(℩xφ)}))
91, 8alrimi 1765 . 2 (∃!xφx(x {x φ} ↔ x {(℩xφ)}))
10 nfab1 2491 . . 3 x{x φ}
11 nfiota1 4341 . . . 4 x(℩xφ)
1211nfsn 3784 . . 3 x{(℩xφ)}
1310, 12cleqf 2513 . 2 ({x φ} = {(℩xφ)} ↔ x(x {x φ} ↔ x {(℩xφ)}))
149, 13sylibr 203 1 (∃!xφ → {x φ} = {(℩xφ)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3737  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by: (None)
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