Theorem abvor0 3568

Description: The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.)

Assertion

Ref	Expression
abvor0	|- ({x | ph} = _V \/ {x | ph} = (/))

Distinct variable group:   ph,x

Proof of Theorem abvor0

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- (ph -> ph)
2		vex 2863	. . . . . . 7 |- x e. _V
3	2	a1i 10	. . . . . 6 |- (ph -> x e. _V)
4	1, 3	2thd 231	. . . . 5 |- (ph -> (ph <-> x e. _V))
5	4	abbi1dv 2470	. . . 4 |- (ph -> {x | ph} = _V)
6	5	con3i 127	. . 3 |- (-. {x | ph} = _V -> -. ph)
7		id 19	. . . . 5 |- (-. ph -> -. ph)
8		noel 3555	. . . . . 6 |- -. x e. (/)
9	8	a1i 10	. . . . 5 |- (-. ph -> -. x e. (/))
10	7, 9	2falsed 340	. . . 4 |- (-. ph -> (ph <-> x e. (/)))
11	10	abbi1dv 2470	. . 3 |- (-. ph -> {x | ph} = (/))
12	6, 11	syl 15	. 2 |- (-. {x | ph} = _V -> {x | ph} = (/))
13	12	orri 365	1 |- ({x | ph} = _V \/ {x | ph} = (/))

Syntax hints:  -. wn 3   \/ wo 357   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2860  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  abexv  4325