Theorem abvor0dc 3488

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 837	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		id 19	. . . . 5 ⊢ (𝜑 → 𝜑)
3		vex 2776	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ V)
5	2, 4	2thd 175	. . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝑥 ∈ V))
6	5	abbi1dv 2326	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
7		id 19	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝜑)
8		noel 3468	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
9	8	a1i 9	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝑥 ∈ ∅)
10	7, 9	2falsed 704	. . . 4 ⊢ (¬ 𝜑 → (𝜑 ↔ 𝑥 ∈ ∅))
11	10	abbi1dv 2326	. . 3 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
12	6, 11	orim12i 761	. 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 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177  {cab 2192  Vcvv 2773  ∅c0 3464

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-nul 3465

This theorem is referenced by: (None)