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Theorem sniota 5187
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 2030 . . 3 𝑥∃!𝑥𝜑
2 iota1 5172 . . . . 5 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3 eqcom 2172 . . . . 5 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
42, 3bitrdi 195 . . . 4 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
5 abid 2158 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
6 vex 2733 . . . . 5 𝑥 ∈ V
76elsn 3597 . . . 4 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
84, 5, 73bitr4g 222 . . 3 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
91, 8alrimi 1515 . 2 (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10 nfab1 2314 . . 3 𝑥{𝑥𝜑}
11 nfiota1 5160 . . . 4 𝑥(℩𝑥𝜑)
1211nfsn 3641 . . 3 𝑥{(℩𝑥𝜑)}
1310, 12cleqf 2337 . 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 1346   = wceq 1348  ∃!weu 2019  wcel 2141  {cab 2156  {csn 3581  cio 5156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-uni 3795  df-iota 5158
This theorem is referenced by:  snriota  5835
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