Theorem dcextest 4557

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 826	. . . 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 4113	. . . . . . 7 |- -. _V e. _V
5		df-v 2727	. . . . . . . . 9 |- _V = { x | x = x }
6		equid 1689	. . . . . . . . . . 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 2283	. . . . . . . . 9 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2211	. . . . . . . 8 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2234	. . . . . . 7 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 665	. . . . . 6 |- ( ph -> -. { x | ph } e. _V )
13	12	con2i 617	. . . . 5 |- ( { x | ph } e. _V -> -. ph )
14		vex 2728	. . . . . . . . . 10 |- y e. _V
15		biidd 171	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
16	14, 15	elab 2869	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
17	16	notbii 658	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
18	17	biimpri 132	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
19	18	eq0rdv 3452	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
20		0ex 4108	. . . . . 6 |- (/) e. _V
21	19, 20	eqeltrdi 2256	. . . . 5 |- ( -. ph -> { x | ph } e. _V )
22	13, 21	impbii 125	. . . 4 |- ( { x | ph } e. _V <-> -. ph )
23	22	notbii 658	. . . 4 |- ( -. { x | ph } e. _V <-> -. -. ph )
24	22, 23	orbi12i 754	. . 3 |- ( ( { x | ph } e. _V \/ -. { x | ph } e. _V ) <-> ( -. ph \/ -. -. ph ) )
25	3, 24	mpbi 144	. 2 |- ( -. ph \/ -. -. ph )
26		df-dc 825	. 2 |- (DECID -. ph <-> ( -. ph \/ -. -. ph ) )
27	25, 26	mpbir 145	1 |- DECID -. ph

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 698  DECID wdc 824   e. 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)