Theorem ordtriexmidlem2 4246

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4247 or weak linearity in ordsoexmid 4286) 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 3228	. . 3 |- -. (/) e. (/)
2		eleq2 2101	. . 3 |- ( { x e. { (/) } | ph } = (/) -> ( (/) e. { x e. { (/) } | ph } <-> (/) e. (/) ) )
3	1, 2	mtbiri 600	. 2 |- ( { x e. { (/) } | ph } = (/) -> -. (/) e. { x e. { (/) } | ph } )
4		0ex 3884	. . . 4 |- (/) e. _V
5	4	snid 3402	. . 3 |- (/) e. { (/) }
6		biidd 161	. . . 4 |- ( x = (/) -> ( ph <-> ph ) )
7	6	elrab3 2699	. . 3 |- ( (/) e. { (/) } -> ( (/) e. { x e. { (/) } | ph } <-> ph ) )
8	5, 7	ax-mp 7	. 2 |- ( (/) e. { x e. { (/) } | ph } <-> ph )
9	3, 8	sylnib 601	1 |- ( { x e. { (/) } | ph } = (/) -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   {crab 2310   (/)c0 3224   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-dif 2920  df-nul 3225  df-sn 3381

This theorem is referenced by:  ordtriexmid  4247  ordtri2orexmid  4248  ontr2exmid  4250  onsucsssucexmid  4252  ordsoexmid  4286  0elsucexmid  4289  ordpwsucexmid  4294