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Theorem abvor0dc 3381
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 820 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 id 19 . . . . 5 (𝜑𝜑)
3 vex 2684 . . . . . 6 𝑥 ∈ V
43a1i 9 . . . . 5 (𝜑𝑥 ∈ V)
52, 42thd 174 . . . 4 (𝜑 → (𝜑𝑥 ∈ V))
65abbi1dv 2257 . . 3 (𝜑 → {𝑥𝜑} = V)
7 id 19 . . . . 5 𝜑 → ¬ 𝜑)
8 noel 3362 . . . . . 6 ¬ 𝑥 ∈ ∅
98a1i 9 . . . . 5 𝜑 → ¬ 𝑥 ∈ ∅)
107, 92falsed 691 . . . 4 𝜑 → (𝜑𝑥 ∈ ∅))
1110abbi1dv 2257 . . 3 𝜑 → {𝑥𝜑} = ∅)
126, 11orim12i 748 . 2 ((𝜑 ∨ ¬ 𝜑) → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
131, 12sylbi 120 1 (DECID 𝜑 → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  {cab 2123  Vcvv 2681  c0 3358
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359
This theorem is referenced by: (None)
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