Theorem abvor0dc 3492

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 837	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		id 19	. . . . 5 |- ( ph -> ph )
3		vex 2779	. . . . . 6 |- x e. _V
4	3	a1i 9	. . . . 5 |- ( ph -> x e. _V )
5	2, 4	2thd 175	. . . 4 |- ( ph -> ( ph <-> x e. _V ) )
6	5	abbi1dv 2327	. . 3 |- ( ph -> { x | ph } = _V )
7		id 19	. . . . 5 |- ( -. ph -> -. ph )
8		noel 3472	. . . . . 6 |- -. x e. (/)
9	8	a1i 9	. . . . 5 |- ( -. ph -> -. x e. (/) )
10	7, 9	2falsed 704	. . . 4 |- ( -. ph -> ( ph <-> x e. (/) ) )
11	10	abbi1dv 2327	. . 3 |- ( -. ph -> { x | ph } = (/) )
12	6, 11	orim12i 761	. 2 |- ( ( ph \/ -. ph ) -> ( { x | ph } = _V \/ { x | ph } = (/) ) )
13	1, 12	sylbi 121	1 |- (DECID ph -> ( { x | ph } = _V \/ { x | ph } = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   (/)c0 3468

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-nul 3469

This theorem is referenced by: (None)