Theorem abvor0dc 3381

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 ph -> ( { x | ph } = _V \/ { x | ph } = (/) ) )

Distinct variable group:   ph, x

Proof of Theorem abvor0dc

Step	Hyp	Ref	Expression
1		df-dc 820	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		id 19	. . . . 5 |- ( ph -> ph )
3		vex 2684	. . . . . 6 |- x e. _V
4	3	a1i 9	. . . . 5 |- ( ph -> x e. _V )
5	2, 4	2thd 174	. . . 4 |- ( ph -> ( ph <-> x e. _V ) )
6	5	abbi1dv 2257	. . 3 |- ( ph -> { x | ph } = _V )
7		id 19	. . . . 5 |- ( -. ph -> -. ph )
8		noel 3362	. . . . . 6 |- -. x e. (/)
9	8	a1i 9	. . . . 5 |- ( -. ph -> -. x e. (/) )
10	7, 9	2falsed 691	. . . 4 |- ( -. ph -> ( ph <-> x e. (/) ) )
11	10	abbi1dv 2257	. . 3 |- ( -. ph -> { x | ph } = (/) )
12	6, 11	orim12i 748	. 2 |- ( ( ph \/ -. ph ) -> ( { x | ph } = _V \/ { x | ph } = (/) ) )
13	1, 12	sylbi 120	1 |- (DECID ph -> ( { x | ph } = _V \/ { x | ph } = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   {cab 2123   _Vcvv 2681   (/)c0 3358

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359

This theorem is referenced by: (None)