Theorem abexv 4325

Description: When x does not occur in ph, {x | ph} is a set. (Contributed by SF, 17-Jan-2015.)

Assertion

Ref	Expression
abexv	|- {x | ph} e. _V

Distinct variable group:   ph,x

Proof of Theorem abexv

Step	Hyp	Ref	Expression
1		abvor0 3568	. 2 |- ({x | ph} = _V \/ {x | ph} = (/))
2		vvex 4110	. . . 4 |- _V e. _V
3		eleq1 2413	. . . 4 |- ({x | ph} = _V -> ({x | ph} e. _V <-> _V e. _V))
4	2, 3	mpbiri 224	. . 3 |- ({x | ph} = _V -> {x | ph} e. _V)
5		0ex 4111	. . . 4 |- (/) e. _V
6		eleq1 2413	. . . 4 |- ({x | ph} = (/) -> ({x | ph} e. _V <-> (/) e. _V))
7	5, 6	mpbiri 224	. . 3 |- ({x | ph} = (/) -> {x | ph} e. _V)
8	4, 7	jaoi 368	. 2 |- (({x | ph} = _V \/ {x | ph} = (/)) -> {x | ph} e. _V)
9	1, 8	ax-mp 5	1 |- {x | ph} e. _V

Syntax hints:   \/ 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  ax-nin 4079

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-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  nncaddccl  4420  preaddccan2lem1  4455  ltfintrilem1  4466  leconnnc  6219  addccan2nclem2  6265  nchoicelem16  6305