Theorem ordtriexmidlem 4245

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4247 or weak linearity in ordsoexmid 4286) 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 102	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. z )
2		elrabi 2695	. . . . . . . . 9 |- ( z e. { x e. { (/) } | ph } -> z e. { (/) } )
3		velsn 3392	. . . . . . . . 9 |- ( z e. { (/) } <-> z = (/) )
4	2, 3	sylib 127	. . . . . . . 8 |- ( z e. { x e. { (/) } | ph } -> z = (/) )
5		noel 3228	. . . . . . . . 9 |- -. y e. (/)
6		eleq2 2101	. . . . . . . . 9 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
7	5, 6	mtbiri 600	. . . . . . . 8 |- ( z = (/) -> -. y e. z )
8	4, 7	syl 14	. . . . . . 7 |- ( z e. { x e. { (/) } | ph } -> -. y e. z )
9	8	adantl 262	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> -. y e. z )
10	1, 9	pm2.21dd 550	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
11	10	gen2 1339	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
12		dftr2 3856	. . . 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 134	. . 3 |- Tr { x e. { (/) } | ph }
14		ssrab2 3025	. . 3 |- { x e. { (/) } | ph } C_ { (/) }
15		ord0 4128	. . . . 5 |- Ord (/)
16		ordsucim 4226	. . . . 5 |- ( Ord (/) -> Ord suc (/) )
17	15, 16	ax-mp 7	. . . 4 |- Ord suc (/)
18		suc0 4148	. . . . 5 |- suc (/) = { (/) }
19		ordeq 4109	. . . . 5 |- ( suc (/) = { (/) } -> ( Ord suc (/) <-> Ord { (/) } ) )
20	18, 19	ax-mp 7	. . . 4 |- ( Ord suc (/) <-> Ord { (/) } )
21	17, 20	mpbi 133	. . 3 |- Ord { (/) }
22		trssord 4117	. . 3 |- ( ( Tr { x e. { (/) } | ph } /\ { x e. { (/) } | ph } C_ { (/) } /\ Ord { (/) } ) -> Ord { x e. { (/) } | ph } )
23	13, 14, 21, 22	mp3an 1232	. 2 |- Ord { x e. { (/) } | ph }
24		p0ex 3939	. . . 4 |- { (/) } e. _V
25	24	rabex 3901	. . 3 |- { x e. { (/) } | ph } e. _V
26	25	elon 4111	. 2 |- ( { x e. { (/) } | ph } e. On <-> Ord { x e. { (/) } | ph } )
27	23, 26	mpbir 134	1 |- { x e. { (/) } | ph } e. On

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {crab 2310   C_ wss 2917   (/)c0 3224   {csn 3375   Tr wtr 3854   Ord word 4099   Oncon0 4100   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by:  ordtriexmid  4247  ordtri2orexmid  4248  ontr2exmid  4250  onsucsssucexmid  4252  ordsoexmid  4286  0elsucexmid  4289  ordpwsucexmid  4294