Theorem abvor0dc 3545

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 847	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		id 19	. . . . 5 |- ( ph -> ph )
3		vex 2824	. . . . . 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 2360	. . 3 |- ( ph -> { x | ph } = _V )
7		id 19	. . . . 5 |- ( -. ph -> -. ph )
8		noel 3525	. . . . . 6 |- -. x e. (/)
9	8	a1i 9	. . . . 5 |- ( -. ph -> -. x e. (/) )
10	7, 9	2falsed 714	. . . 4 |- ( -. ph -> ( ph <-> x e. (/) ) )
11	10	abbi1dv 2360	. . 3 |- ( -. ph -> { x | ph } = (/) )
12	6, 11	orim12i 771	. 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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521

This theorem is referenced by: (None)