Theorem abvor0dc 3269

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 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))

Distinct variable group:   𝜑,𝑥

Proof of Theorem abvor0dc

Step	Hyp	Ref	Expression
1		df-dc 776	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		id 19	. . . . 5 ⊢ (𝜑 → 𝜑)
3		vex 2604	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ V)
5	2, 4	2thd 173	. . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝑥 ∈ V))
6	5	abbi1dv 2198	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
7		id 19	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝜑)
8		noel 3255	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
9	8	a1i 9	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝑥 ∈ ∅)
10	7, 9	2falsed 650	. . . 4 ⊢ (¬ 𝜑 → (𝜑 ↔ 𝑥 ∈ ∅))
11	10	abbi1dv 2198	. . 3 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
12	6, 11	orim12i 708	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
13	1, 12	sylbi 119	1 ⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  DECID wdc 775   = wceq 1284   ∈ wcel 1433  {cab 2067  Vcvv 2601  ∅c0 3251

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by: (None)