Theorem ordtriexmidlem2 4521

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4522 or weak linearity in ordsoexmid 4563) 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 3428	. . 3 |- -. (/) e. (/)
2		eleq2 2241	. . 3 |- ( { x e. { (/) } | ph } = (/) -> ( (/) e. { x e. { (/) } | ph } <-> (/) e. (/) ) )
3	1, 2	mtbiri 675	. 2 |- ( { x e. { (/) } | ph } = (/) -> -. (/) e. { x e. { (/) } | ph } )
4		0ex 4132	. . . 4 |- (/) e. _V
5	4	snid 3625	. . 3 |- (/) e. { (/) }
6		biidd 172	. . . 4 |- ( x = (/) -> ( ph <-> ph ) )
7	6	elrab3 2896	. . 3 |- ( (/) e. { (/) } -> ( (/) e. { x e. { (/) } | ph } <-> ph ) )
8	5, 7	ax-mp 5	. 2 |- ( (/) e. { x e. { (/) } | ph } <-> ph )
9	3, 8	sylnib 676	1 |- ( { x e. { (/) } | ph } = (/) -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   {crab 2459   (/)c0 3424   {csn 3594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4131

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-nul 3425  df-sn 3600

This theorem is referenced by:  ordtriexmid  4522  ontriexmidim  4523  ordtri2orexmid  4524  ontr2exmid  4526  onsucsssucexmid  4528  ordsoexmid  4563  0elsucexmid  4566  ordpwsucexmid  4571