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Theorem abvor0dc 3276
Description: The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
abvor0dc (DECID 𝜑 → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
Distinct variable group:   𝜑,𝑥

Proof of Theorem abvor0dc
StepHypRef Expression
1 df-dc 777 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 id 19 . . . . 5 (𝜑𝜑)
3 vex 2605 . . . . . 6 𝑥 ∈ V
43a1i 9 . . . . 5 (𝜑𝑥 ∈ V)
52, 42thd 173 . . . 4 (𝜑 → (𝜑𝑥 ∈ V))
65abbi1dv 2199 . . 3 (𝜑 → {𝑥𝜑} = V)
7 id 19 . . . . 5 𝜑 → ¬ 𝜑)
8 noel 3262 . . . . . 6 ¬ 𝑥 ∈ ∅
98a1i 9 . . . . 5 𝜑 → ¬ 𝑥 ∈ ∅)
107, 92falsed 651 . . . 4 𝜑 → (𝜑𝑥 ∈ ∅))
1110abbi1dv 2199 . . 3 𝜑 → {𝑥𝜑} = ∅)
126, 11orim12i 709 . 2 ((𝜑 ∨ ¬ 𝜑) → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
131, 12sylbi 119 1 (DECID 𝜑 → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  DECID wdc 776   = wceq 1285  wcel 1434  {cab 2068  Vcvv 2602  c0 3258
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-in1 577  ax-in2 578  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 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-nul 3259
This theorem is referenced by: (None)
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