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Theorem abvor0dc 3242
 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 743 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 id 19 . . . . 5 (𝜑𝜑)
3 vex 2560 . . . . . 6 𝑥 ∈ V
43a1i 9 . . . . 5 (𝜑𝑥 ∈ V)
52, 42thd 164 . . . 4 (𝜑 → (𝜑𝑥 ∈ V))
65abbi1dv 2157 . . 3 (𝜑 → {𝑥𝜑} = V)
7 id 19 . . . . 5 𝜑 → ¬ 𝜑)
8 noel 3228 . . . . . 6 ¬ 𝑥 ∈ ∅
98a1i 9 . . . . 5 𝜑 → ¬ 𝑥 ∈ ∅)
107, 92falsed 618 . . . 4 𝜑 → (𝜑𝑥 ∈ ∅))
1110abbi1dv 2157 . . 3 𝜑 → {𝑥𝜑} = ∅)
126, 11orim12i 676 . 2 ((𝜑 ∨ ¬ 𝜑) → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
131, 12sylbi 114 1 (DECID 𝜑 → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742   = wceq 1243   ∈ wcel 1393  {cab 2026  Vcvv 2557  ∅c0 3224 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-in1 544  ax-in2 545  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-dc 743  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225 This theorem is referenced by: (None)
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