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Theorem sniota 5020
 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 1960 . . 3 𝑥∃!𝑥𝜑
2 iota1 5007 . . . . 5 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3 eqcom 2091 . . . . 5 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
42, 3syl6bb 195 . . . 4 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
5 abid 2077 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
6 vex 2623 . . . . 5 𝑥 ∈ V
76elsn 3466 . . . 4 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
84, 5, 73bitr4g 222 . . 3 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
91, 8alrimi 1461 . 2 (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10 nfab1 2231 . . 3 𝑥{𝑥𝜑}
11 nfiota1 4995 . . . 4 𝑥(℩𝑥𝜑)
1211nfsn 3506 . . 3 𝑥{(℩𝑥𝜑)}
1310, 12cleqf 2253 . 2 ({𝑥𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
149, 13sylibr 133 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1288   = wceq 1290   ∈ wcel 1439  ∃!weu 1949  {cab 2075  {csn 3450  ℩cio 4991 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-sn 3456  df-pr 3457  df-uni 3660  df-iota 4993 This theorem is referenced by:  snriota  5651
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