Theorem abvor0dc 3432

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 825	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		id 19	. . . . 5 ⊢ (𝜑 → 𝜑)
3		vex 2729	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ V)
5	2, 4	2thd 174	. . . 4 ⊢ (𝜑 → (𝜑 ↔ 𝑥 ∈ V))
6	5	abbi1dv 2286	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
7		id 19	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝜑)
8		noel 3413	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
9	8	a1i 9	. . . . 5 ⊢ (¬ 𝜑 → ¬ 𝑥 ∈ ∅)
10	7, 9	2falsed 692	. . . 4 ⊢ (¬ 𝜑 → (𝜑 ↔ 𝑥 ∈ ∅))
11	10	abbi1dv 2286	. . 3 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
12	6, 11	orim12i 749	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
13	1, 12	sylbi 120	1 ⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 824   = wceq 1343   ∈ wcel 2136  {cab 2151  Vcvv 2726  ∅c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by: (None)