Theorem ordtriexmidlem 4661

Description: Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4663 or weak linearity in ordsoexmid 4704) 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 2979	. . . . . . . . 9 |- ( z e. { x e. { (/) } | ph } -> z e. { (/) } )
3		velsn 3722	. . . . . . . . 9 |- ( z e. { (/) } <-> z = (/) )
4	2, 3	sylib 122	. . . . . . . 8 |- ( z e. { x e. { (/) } | ph } -> z = (/) )
5		noel 3525	. . . . . . . . 9 |- -. y e. (/)
6		eleq2 2302	. . . . . . . . 9 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
7	5, 6	mtbiri 686	. . . . . . . 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 629	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
11	10	gen2 1503	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) } | ph } ) -> y e. { x e. { (/) } | ph } )
12		dftr2 4226	. . . 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 3333	. . 3 |- { x e. { (/) } | ph } C_ { (/) }
15		ord0 4531	. . . . 5 |- Ord (/)
16		ordsucim 4642	. . . . 5 |- ( Ord (/) -> Ord suc (/) )
17	15, 16	ax-mp 5	. . . 4 |- Ord suc (/)
18		suc0 4551	. . . . 5 |- suc (/) = { (/) }
19		ordeq 4512	. . . . 5 |- ( suc (/) = { (/) } -> ( Ord suc (/) <-> Ord { (/) } ) )
20	18, 19	ax-mp 5	. . . 4 |- ( Ord suc (/) <-> Ord { (/) } )
21	17, 20	mpbi 145	. . 3 |- Ord { (/) }
22		trssord 4520	. . 3 |- ( ( Tr { x e. { (/) } | ph } /\ { x e. { (/) } | ph } C_ { (/) } /\ Ord { (/) } ) -> Ord { x e. { (/) } | ph } )
23	13, 14, 21, 22	mp3an 1378	. 2 |- Ord { x e. { (/) } | ph }
24		p0ex 4320	. . . 4 |- { (/) } e. _V
25	24	rabex 4275	. . 3 |- { x e. { (/) } | ph } e. _V
26	25	elon 4514	. 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 1400   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   (/)c0 3520   {csn 3705   Tr wtr 4224   Ord word 4502   Oncon0 4503   suc csuc 4505

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by:  ordtriexmid  4663  ontriexmidim  4664  ordtri2orexmid  4665  ontr2exmid  4667  onsucsssucexmid  4669  ordsoexmid  4704  0elsucexmid  4707  ordpwsucexmid  4712  unfiexmid  7215  exmidonfinlem  7535