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Theorem sniota 4945
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 1954 . . 3 𝑥∃!𝑥𝜑
2 iota1 4932 . . . . 5 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3 eqcom 2085 . . . . 5 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
42, 3syl6bb 194 . . . 4 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
5 abid 2071 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
6 vex 2614 . . . . 5 𝑥 ∈ V
76elsn 3433 . . . 4 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
84, 5, 73bitr4g 221 . . 3 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
91, 8alrimi 1456 . 2 (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10 nfab1 2225 . . 3 𝑥{𝑥𝜑}
11 nfiota1 4920 . . . 4 𝑥(℩𝑥𝜑)
1211nfsn 3471 . . 3 𝑥{(℩𝑥𝜑)}
1310, 12cleqf 2246 . 2 ({𝑥𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
149, 13sylibr 132 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283   = wceq 1285  wcel 1434  ∃!weu 1943  {cab 2069  {csn 3417  cio 4916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2613  df-sbc 2826  df-un 2987  df-sn 3423  df-pr 3424  df-uni 3623  df-iota 4918
This theorem is referenced by:  snriota  5550
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