Theorem dcextest 4557

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 826	. . . 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 4113	. . . . . . 7 ⊢ ¬ V ∈ V
5		df-v 2727	. . . . . . . . 9 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1689	. . . . . . . . . . 11 ⊢ 𝑥 = 𝑥
7		pm5.1im 172	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥 ↔ 𝜑)))
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = 𝑥 ↔ 𝜑))
9	8	abbidv 2283	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	eqtr2id 2211	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2234	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 665	. . . . . 6 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13	12	con2i 617	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} ∈ V → ¬ 𝜑)
14		vex 2728	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15		biidd 171	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
16	14, 15	elab 2869	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
17	16	notbii 658	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
18	17	biimpri 132	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
19	18	eq0rdv 3452	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
20		0ex 4108	. . . . . 6 ⊢ ∅ ∈ V
21	19, 20	eqeltrdi 2256	. . . . 5 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} ∈ V)
22	13, 21	impbii 125	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ¬ 𝜑)
23	22	notbii 658	. . . 4 ⊢ (¬ {𝑥 ∣ 𝜑} ∈ V ↔ ¬ ¬ 𝜑)
24	22, 23	orbi12i 754	. . 3 ⊢ (({𝑥 ∣ 𝜑} ∈ V ∨ ¬ {𝑥 ∣ 𝜑} ∈ V) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
25	3, 24	mpbi 144	. 2 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)
26		df-dc 825	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
27	25, 26	mpbir 145	1 ⊢ DECID ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 698  DECID wdc 824   ∈ wcel 2136  {cab 2151  Vcvv 2725  ∅c0 3408

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-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-nul 4107

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-dif 3117  df-in 3121  df-ss 3128  df-nul 3409

This theorem is referenced by: (None)