Theorem ordtriexmidlem2 4504

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4505 or weak linearity in ordsoexmid 4546) 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 3418	. . 3 |- -. (/) e. (/)
2		eleq2 2234	. . 3 |- ( { x e. { (/) } | ph } = (/) -> ( (/) e. { x e. { (/) } | ph } <-> (/) e. (/) ) )
3	1, 2	mtbiri 670	. 2 |- ( { x e. { (/) } | ph } = (/) -> -. (/) e. { x e. { (/) } | ph } )
4		0ex 4116	. . . 4 |- (/) e. _V
5	4	snid 3614	. . 3 |- (/) e. { (/) }
6		biidd 171	. . . 4 |- ( x = (/) -> ( ph <-> ph ) )
7	6	elrab3 2887	. . 3 |- ( (/) e. { (/) } -> ( (/) e. { x e. { (/) } | ph } <-> ph ) )
8	5, 7	ax-mp 5	. 2 |- ( (/) e. { x e. { (/) } | ph } <-> ph )
9	3, 8	sylnib 671	1 |- ( { x e. { (/) } | ph } = (/) -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452   (/)c0 3414   {csn 3583

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3589

This theorem is referenced by:  ordtriexmid  4505  ontriexmidim  4506  ordtri2orexmid  4507  ontr2exmid  4509  onsucsssucexmid  4511  ordsoexmid  4546  0elsucexmid  4549  ordpwsucexmid  4554