Theorem abvor0dc 3438

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 830	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		id 19	. . . . 5 |- ( ph -> ph )
3		vex 2733	. . . . . 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 2290	. . 3 |- ( ph -> { x | ph } = _V )
7		id 19	. . . . 5 |- ( -. ph -> -. ph )
8		noel 3418	. . . . . 6 |- -. x e. (/)
9	8	a1i 9	. . . . 5 |- ( -. ph -> -. x e. (/) )
10	7, 9	2falsed 697	. . . 4 |- ( -. ph -> ( ph <-> x e. (/) ) )
11	10	abbi1dv 2290	. . 3 |- ( -. ph -> { x | ph } = (/) )
12	6, 11	orim12i 754	. 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 703  DECID wdc 829   = wceq 1348   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-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  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-nul 3415

This theorem is referenced by: (None)