Theorem dcextest 4565

Description: If it is decidable whether { x | ph } is a set, then -. ph is decidable (where x does not occur in ph). 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 -. ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)

Hypothesis

Ref	Expression
dcextest.ex	|- DECID { x | ph } e. _V

Assertion

Ref	Expression
dcextest	|- DECID -. ph

Distinct variable group:   ph, x

Proof of Theorem dcextest

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dcextest.ex	. . . 4 |- DECID { x | ph } e. _V
2		exmiddc 831	. . . 4 |- (DECID { x | ph } e. _V -> ( { x | ph } e. _V \/ -. { x | ph } e. _V ) )
3	1, 2	ax-mp 5	. . 3 |- ( { x | ph } e. _V \/ -. { x | ph } e. _V )
4		vprc 4121	. . . . . . 7 |- -. _V e. _V
5		df-v 2732	. . . . . . . . 9 |- _V = { x | x = x }
6		equid 1694	. . . . . . . . . . 11 |- x = x
7		pm5.1im 172	. . . . . . . . . . 11 |- ( x = x -> ( ph -> ( x = x <-> ph ) ) )
8	6, 7	ax-mp 5	. . . . . . . . . 10 |- ( ph -> ( x = x <-> ph ) )
9	8	abbidv 2288	. . . . . . . . 9 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2216	. . . . . . . 8 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2239	. . . . . . 7 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 670	. . . . . 6 |- ( ph -> -. { x | ph } e. _V )
13	12	con2i 622	. . . . 5 |- ( { x | ph } e. _V -> -. ph )
14		vex 2733	. . . . . . . . . 10 |- y e. _V
15		biidd 171	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
16	14, 15	elab 2874	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
17	16	notbii 663	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
18	17	biimpri 132	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
19	18	eq0rdv 3459	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
20		0ex 4116	. . . . . 6 |- (/) e. _V
21	19, 20	eqeltrdi 2261	. . . . 5 |- ( -. ph -> { x | ph } e. _V )
22	13, 21	impbii 125	. . . 4 |- ( { x | ph } e. _V <-> -. ph )
23	22	notbii 663	. . . 4 |- ( -. { x | ph } e. _V <-> -. -. ph )
24	22, 23	orbi12i 759	. . 3 |- ( ( { x | ph } e. _V \/ -. { x | ph } e. _V ) <-> ( -. ph \/ -. -. ph ) )
25	3, 24	mpbi 144	. 2 |- ( -. ph \/ -. -. ph )
26		df-dc 830	. 2 |- (DECID -. ph <-> ( -. ph \/ -. -. ph ) )
27	25, 26	mpbir 145	1 |- DECID -. ph

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 703  DECID wdc 829   e. wcel 2141   {cab 2156   _Vcvv 2730   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by: (None)