Theorem ordtriexmidlem 4641

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4643 or weak linearity in ordsoexmid 4684) 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 2970	. . . . . . . . 9 |- ( z e. { x e. { (/) } | ph } -> z e. { (/) } )
3		velsn 3706	. . . . . . . . 9 |- ( z e. { (/) } <-> z = (/) )
4	2, 3	sylib 122	. . . . . . . 8 |- ( z e. { x e. { (/) } | ph } -> z = (/) )
5		noel 3512	. . . . . . . . 9 |- -. y e. (/)
6		eleq2 2296	. . . . . . . . 9 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
7	5, 6	mtbiri 682	. . . . . . . 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 625	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
11	10	gen2 1499	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
12		dftr2 4210	. . . 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 3323	. . 3 |- { x e. { (/) } | ph } C_ { (/) }
15		ord0 4512	. . . . 5 |- Ord (/)
16		ordsucim 4622	. . . . 5 |- ( Ord (/) -> Ord suc (/) )
17	15, 16	ax-mp 5	. . . 4 |- Ord suc (/)
18		suc0 4532	. . . . 5 |- suc (/) = { (/) }
19		ordeq 4493	. . . . 5 |- ( suc (/) = { (/) } -> ( Ord suc (/) <-> Ord { (/) } ) )
20	18, 19	ax-mp 5	. . . 4 |- ( Ord suc (/) <-> Ord { (/) } )
21	17, 20	mpbi 145	. . 3 |- Ord { (/) }
22		trssord 4501	. . 3 |- ( ( Tr { x e. { (/) } | ph } /\ { x e. { (/) } | ph } C_ { (/) } /\ Ord { (/) } ) -> Ord { x e. { (/) } | ph } )
23	13, 14, 21, 22	mp3an 1374	. 2 |- Ord { x e. { (/) } | ph }
24		p0ex 4301	. . . 4 |- { (/) } e. _V
25	24	rabex 4256	. . 3 |- { x e. { (/) } | ph } e. _V
26	25	elon 4495	. 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 1396   = wceq 1398   e. wcel 2203   {crab 2524   C_ wss 3211   (/)c0 3508   {csn 3689   Tr wtr 4208   Ord word 4483   Oncon0 4484   suc csuc 4486

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by:  ordtriexmid  4643  ontriexmidim  4644  ordtri2orexmid  4645  ontr2exmid  4647  onsucsssucexmid  4649  ordsoexmid  4684  0elsucexmid  4687  ordpwsucexmid  4692  unfiexmid  7178  exmidonfinlem  7496