Theorem abvor0dc 3461

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 836	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		id 19	. . . . 5 ⊢ (𝜑 → 𝜑)
3		vex 2755	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ V)
5	2, 4	2thd 175	. . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝑥 ∈ V))
6	5	abbi1dv 2309	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
7		id 19	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝜑)
8		noel 3441	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
9	8	a1i 9	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝑥 ∈ ∅)
10	7, 9	2falsed 703	. . . 4 ⊢ (¬ 𝜑 → (𝜑 ↔ 𝑥 ∈ ∅))
11	10	abbi1dv 2309	. . 3 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
12	6, 11	orim12i 760	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
13	1, 12	sylbi 121	1 ⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2160  {cab 2175  Vcvv 2752  ∅c0 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438

This theorem is referenced by: (None)