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Theorem abvor0dc 3236
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 742 . 2 (DECID φ ↔ (φ ¬ φ))
2 id 19 . . . . 5 (φφ)
3 vex 2554 . . . . . 6 x V
43a1i 9 . . . . 5 (φx V)
52, 42thd 164 . . . 4 (φ → (φx V))
65abbi1dv 2154 . . 3 (φ → {xφ} = V)
7 id 19 . . . . 5 φ → ¬ φ)
8 noel 3222 . . . . . 6 ¬ x
98a1i 9 . . . . 5 φ → ¬ x ∅)
107, 92falsed 617 . . . 4 φ → (φx ∅))
1110abbi1dv 2154 . . 3 φ → {xφ} = ∅)
126, 11orim12i 675 . 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 628  DECID wdc 741   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551  c0 3218
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219
This theorem is referenced by: (None)
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