Theorem dcextest 4617

Description: If it is decidable whether {𝑥 ∣ 𝜑} is a set, then ¬ 𝜑 is decidable (where 𝑥 does not occur in 𝜑). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition ¬ 𝜑 is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)

Hypothesis

Ref	Expression
dcextest.ex	⊢ DECID {𝑥 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
dcextest	⊢ DECID ¬ 𝜑

Distinct variable group:   𝜑,𝑥

Proof of Theorem dcextest

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dcextest.ex	. . . 4 ⊢ DECID {𝑥 ∣ 𝜑} ∈ V
2		exmiddc 837	. . . 4 ⊢ (DECID {𝑥 ∣ 𝜑} ∈ V → ({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V))
3	1, 2	ax-mp 5	. . 3 ⊢ ({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V)
4		vprc 4165	. . . . . . 7 ⊢ ¬ V ∈ V
5		df-v 2765	. . . . . . . . 9 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1715	. . . . . . . . . . 11 ⊢ 𝑥 = 𝑥
7		pm5.1im 173	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥 ↔ 𝜑)))
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = 𝑥 ↔ 𝜑))
9	8	abbidv 2314	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	eqtr2id 2242	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2265	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 676	. . . . . 6 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13	12	con2i 628	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} ∈ V → ¬ 𝜑)
14		vex 2766	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15		biidd 172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
16	14, 15	elab 2908	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
17	16	notbii 669	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
18	17	biimpri 133	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
19	18	eq0rdv 3495	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
20		0ex 4160	. . . . . 6 ⊢ ∅ ∈ V
21	19, 20	eqeltrdi 2287	. . . . 5 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} ∈ V)
22	13, 21	impbii 126	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ¬ 𝜑)
23	22	notbii 669	. . . 4 ⊢ (¬ {𝑥 ∣ 𝜑} ∈ V ↔ ¬ ¬ 𝜑)
24	22, 23	orbi12i 765	. . 3 ⊢ (({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
25	3, 24	mpbi 145	. 2 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)
26		df-dc 836	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
27	25, 26	mpbir 146	1 ⊢ DECID ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709  DECID wdc 835   ∈ wcel 2167  {cab 2182  Vcvv 2763  ∅c0 3450

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by: (None)