Theorem ordtriexmidlem 4611

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4613 or weak linearity in ordsoexmid 4654) with a proposition ph. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)

Assertion

Ref	Expression
ordtriexmidlem	|- { x e. { (/) } | ph } e. On

Proof of Theorem ordtriexmidlem

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. z )
2		elrabi 2956	. . . . . . . . 9 |- ( z e. { x e. { (/) } | ph } -> z e. { (/) } )
3		velsn 3683	. . . . . . . . 9 |- ( z e. { (/) } <-> z = (/) )
4	2, 3	sylib 122	. . . . . . . 8 |- ( z e. { x e. { (/) } | ph } -> z = (/) )
5		noel 3495	. . . . . . . . 9 |- -. y e. (/)
6		eleq2 2293	. . . . . . . . 9 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
7	5, 6	mtbiri 679	. . . . . . . 8 |- ( z = (/) -> -. y e. z )
8	4, 7	syl 14	. . . . . . 7 |- ( z e. { x e. { (/) } | ph } -> -. y e. z )
9	8	adantl 277	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> -. y e. z )
10	1, 9	pm2.21dd 623	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
11	10	gen2 1496	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
12		dftr2 4184	. . . 4 |- ( Tr { x e. { (/) } | ph } <-> A. y A. z ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } ) )
13	11, 12	mpbir 146	. . 3 |- Tr { x e. { (/) } | ph }
14		ssrab2 3309	. . 3 |- { x e. { (/) } | ph } C_ { (/) }
15		ord0 4482	. . . . 5 |- Ord (/)
16		ordsucim 4592	. . . . 5 |- ( Ord (/) -> Ord suc (/) )
17	15, 16	ax-mp 5	. . . 4 |- Ord suc (/)
18		suc0 4502	. . . . 5 |- suc (/) = { (/) }
19		ordeq 4463	. . . . 5 |- ( suc (/) = { (/) } -> ( Ord suc (/) <-> Ord { (/) } ) )
20	18, 19	ax-mp 5	. . . 4 |- ( Ord suc (/) <-> Ord { (/) } )
21	17, 20	mpbi 145	. . 3 |- Ord { (/) }
22		trssord 4471	. . 3 |- ( ( Tr { x e. { (/) } | ph } /\ { x e. { (/) } | ph } C_ { (/) } /\ Ord { (/) } ) -> Ord { x e. { (/) } | ph } )
23	13, 14, 21, 22	mp3an 1371	. 2 |- Ord { x e. { (/) } | ph }
24		p0ex 4272	. . . 4 |- { (/) } e. _V
25	24	rabex 4228	. . 3 |- { x e. { (/) } | ph } e. _V
26	25	elon 4465	. 2 |- ( { x e. { (/) } | ph } e. On <-> Ord { x e. { (/) } | ph } )
27	23, 26	mpbir 146	1 |- { x e. { (/) } | ph } e. On

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   (/)c0 3491   {csn 3666   Tr wtr 4182   Ord word 4453   Oncon0 4454   suc csuc 4456

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462

This theorem is referenced by:  ordtriexmid  4613  ontriexmidim  4614  ordtri2orexmid  4615  ontr2exmid  4617  onsucsssucexmid  4619  ordsoexmid  4654  0elsucexmid  4657  ordpwsucexmid  4662  unfiexmid  7080  exmidonfinlem  7371