ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sniota GIF version

Theorem sniota 4894
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 1911 . . 3 𝑥∃!𝑥𝜑
2 iota1 4881 . . . . 5 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3 eqcom 2042 . . . . 5 ((℩𝑥𝜑) = 𝑥𝑥 = (℩𝑥𝜑))
42, 3syl6bb 185 . . . 4 (∃!𝑥𝜑 → (𝜑𝑥 = (℩𝑥𝜑)))
5 abid 2028 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
6 vex 2560 . . . . 5 𝑥 ∈ V
76elsn 3391 . . . 4 (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
84, 5, 73bitr4g 212 . . 3 (∃!𝑥𝜑 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
91, 8alrimi 1415 . 2 (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10 nfab1 2180 . . 3 𝑥{𝑥𝜑}
11 nfiota1 4869 . . . 4 𝑥(℩𝑥𝜑)
1211nfsn 3430 . . 3 𝑥{(℩𝑥𝜑)}
1310, 12cleqf 2201 . 2 ({𝑥𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
149, 13sylibr 137 1 (∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241   = wceq 1243  wcel 1393  ∃!weu 1900  {cab 2026  {csn 3375  cio 4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867
This theorem is referenced by:  snriota  5497
  Copyright terms: Public domain W3C validator