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Theorem sniota 4921
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 1927 . . 3 𝑥∃!𝑥𝜑
2 iota1 4908 . . . . 5 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3 eqcom 2058 . . . . 5 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
42, 3syl6bb 189 . . . 4 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
5 abid 2044 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
6 vex 2577 . . . . 5 𝑥 ∈ V
76elsn 3418 . . . 4 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
84, 5, 73bitr4g 216 . . 3 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
91, 8alrimi 1431 . 2 (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10 nfab1 2196 . . 3 𝑥{𝑥𝜑}
11 nfiota1 4896 . . . 4 𝑥(℩𝑥𝜑)
1211nfsn 3457 . . 3 𝑥{(℩𝑥𝜑)}
1310, 12cleqf 2217 . 2 ({𝑥𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
149, 13sylibr 141 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  ∃!weu 1916  {cab 2042  {csn 3402  cio 4892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-sn 3408  df-pr 3409  df-uni 3608  df-iota 4894
This theorem is referenced by:  snriota  5524
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