Theorem ordtriexmidlem2 4497

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4498 or weak linearity in ordsoexmid 4539) 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 3413	. . 3 |- -. (/) e. (/)
2		eleq2 2230	. . 3 |- ( { x e. { (/) } | ph } = (/) -> ( (/) e. { x e. { (/) } | ph } <-> (/) e. (/) ) )
3	1, 2	mtbiri 665	. 2 |- ( { x e. { (/) } | ph } = (/) -> -. (/) e. { x e. { (/) } | ph } )
4		0ex 4109	. . . 4 |- (/) e. _V
5	4	snid 3607	. . 3 |- (/) e. { (/) }
6		biidd 171	. . . 4 |- ( x = (/) -> ( ph <-> ph ) )
7	6	elrab3 2883	. . 3 |- ( (/) e. { (/) } -> ( (/) e. { x e. { (/) } | ph } <-> ph ) )
8	5, 7	ax-mp 5	. 2 |- ( (/) e. { x e. { (/) } | ph } <-> ph )
9	3, 8	sylnib 666	1 |- ( { x e. { (/) } | ph } = (/) -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   {crab 2448   (/)c0 3409   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118  df-nul 3410  df-sn 3582

This theorem is referenced by:  ordtriexmid  4498  ontriexmidim  4499  ordtri2orexmid  4500  ontr2exmid  4502  onsucsssucexmid  4504  ordsoexmid  4539  0elsucexmid  4542  ordpwsucexmid  4547