Theorem ordtriexmidlem2 4337

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4338 or weak linearity in ordsoexmid 4378) with a proposition ph. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)

Assertion

Ref	Expression
ordtriexmidlem2	|- ( { x e. { (/) } | ph } = (/) -> -. ph )

Distinct variable group:   ph, x

Proof of Theorem ordtriexmidlem2

Step	Hyp	Ref	Expression
1		noel 3290	. . 3 |- -. (/) e. (/)
2		eleq2 2151	. . 3 |- ( { x e. { (/) } | ph } = (/) -> ( (/) e. { x e. { (/) } | ph } <-> (/) e. (/) ) )
3	1, 2	mtbiri 635	. 2 |- ( { x e. { (/) } | ph } = (/) -> -. (/) e. { x e. { (/) } | ph } )
4		0ex 3966	. . . 4 |- (/) e. _V
5	4	snid 3475	. . 3 |- (/) e. { (/) }
6		biidd 170	. . . 4 |- ( x = (/) -> ( ph <-> ph ) )
7	6	elrab3 2772	. . 3 |- ( (/) e. { (/) } -> ( (/) e. { x e. { (/) } | ph } <-> ph ) )
8	5, 7	ax-mp 7	. 2 |- ( (/) e. { x e. { (/) } | ph } <-> ph )
9	3, 8	sylnib 636	1 |- ( { x e. { (/) } | ph } = (/) -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   {crab 2363   (/)c0 3286   {csn 3446

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 3001  df-nul 3287  df-sn 3452

This theorem is referenced by:  ordtriexmid  4338  ordtri2orexmid  4339  ontr2exmid  4341  onsucsssucexmid  4343  ordsoexmid  4378  0elsucexmid  4381  ordpwsucexmid  4386