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Theorem abvor0dc 3267
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 752 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 id 19 . . . . 5 (𝜑𝜑)
3 vex 2575 . . . . . 6 𝑥 ∈ V
43a1i 9 . . . . 5 (𝜑𝑥 ∈ V)
52, 42thd 168 . . . 4 (𝜑 → (𝜑𝑥 ∈ V))
65abbi1dv 2171 . . 3 (𝜑 → {𝑥𝜑} = V)
7 id 19 . . . . 5 𝜑 → ¬ 𝜑)
8 noel 3253 . . . . . 6 ¬ 𝑥 ∈ ∅
98a1i 9 . . . . 5 𝜑 → ¬ 𝑥 ∈ ∅)
107, 92falsed 626 . . . 4 𝜑 → (𝜑𝑥 ∈ ∅))
1110abbi1dv 2171 . . 3 𝜑 → {𝑥𝜑} = ∅)
126, 11orim12i 684 . 2 ((𝜑 ∨ ¬ 𝜑) → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
131, 12sylbi 118 1 (DECID 𝜑 → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 637  DECID wdc 751   = wceq 1257  wcel 1407  {cab 2040  Vcvv 2572  c0 3249
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-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-dc 752  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-dif 2945  df-nul 3250
This theorem is referenced by: (None)
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