Theorem dfom2 7067

Description: An alternate definition of the set of natural numbers om. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol K_I for the inner class builder of non-limit ordinal numbers (see nlimon 7051). (Contributed by NM, 1-Nov-2004.)

Assertion

Ref	Expression
dfom2	|- om = { x e. On | suc x C_ { y e. On | -. Lim y } }

Proof of Theorem dfom2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-om 7066	. 2 |- om = { x e. On | A. z ( Lim z -> x e. z ) }
2		onsssuc 5813	. . . . . . . . . . 11 |- ( ( z e. On /\ x e. On ) -> ( z C_ x <-> z e. suc x ) )
3		ontri1 5757	. . . . . . . . . . 11 |- ( ( z e. On /\ x e. On ) -> ( z C_ x <-> -. x e. z ) )
4	2, 3	bitr3d 270	. . . . . . . . . 10 |- ( ( z e. On /\ x e. On ) -> ( z e. suc x <-> -. x e. z ) )
5	4	ancoms 469	. . . . . . . . 9 |- ( ( x e. On /\ z e. On ) -> ( z e. suc x <-> -. x e. z ) )
6		limeq 5735	. . . . . . . . . . . 12 |- ( y = z -> ( Lim y <-> Lim z ) )
7	6	notbid 308	. . . . . . . . . . 11 |- ( y = z -> ( -. Lim y <-> -. Lim z ) )
8	7	elrab 3363	. . . . . . . . . 10 |- ( z e. { y e. On | -. Lim y } <-> ( z e. On /\ -. Lim z ) )
9	8	a1i 11	. . . . . . . . 9 |- ( ( x e. On /\ z e. On ) -> ( z e. { y e. On | -. Lim y } <-> ( z e. On /\ -. Lim z ) ) )
10	5, 9	imbi12d 334	. . . . . . . 8 |- ( ( x e. On /\ z e. On ) -> ( ( z e. suc x -> z e. { y e. On | -. Lim y } ) <-> ( -. x e. z -> ( z e. On /\ -. Lim z ) ) ) )
11	10	pm5.74da 723	. . . . . . 7 |- ( x e. On -> ( ( z e. On -> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) <-> ( z e. On -> ( -. x e. z -> ( z e. On /\ -. Lim z ) ) ) ) )
12		vex 3203	. . . . . . . . . . 11 |- z e. _V
13		limelon 5788	. . . . . . . . . . 11 |- ( ( z e. _V /\ Lim z ) -> z e. On )
14	12, 13	mpan 706	. . . . . . . . . 10 |- ( Lim z -> z e. On )
15	14	pm4.71ri 665	. . . . . . . . 9 |- ( Lim z <-> ( z e. On /\ Lim z ) )
16	15	imbi1i 339	. . . . . . . 8 |- ( ( Lim z -> x e. z ) <-> ( ( z e. On /\ Lim z ) -> x e. z ) )
17		impexp 462	. . . . . . . 8 |- ( ( ( z e. On /\ Lim z ) -> x e. z ) <-> ( z e. On -> ( Lim z -> x e. z ) ) )
18		con34b 306	. . . . . . . . . 10 |- ( ( Lim z -> x e. z ) <-> ( -. x e. z -> -. Lim z ) )
19		ibar 525	. . . . . . . . . . 11 |- ( z e. On -> ( -. Lim z <-> ( z e. On /\ -. Lim z ) ) )
20	19	imbi2d 330	. . . . . . . . . 10 |- ( z e. On -> ( ( -. x e. z -> -. Lim z ) <-> ( -. x e. z -> ( z e. On /\ -. Lim z ) ) ) )
21	18, 20	syl5bb 272	. . . . . . . . 9 |- ( z e. On -> ( ( Lim z -> x e. z ) <-> ( -. x e. z -> ( z e. On /\ -. Lim z ) ) ) )
22	21	pm5.74i 260	. . . . . . . 8 |- ( ( z e. On -> ( Lim z -> x e. z ) ) <-> ( z e. On -> ( -. x e. z -> ( z e. On /\ -. Lim z ) ) ) )
23	16, 17, 22	3bitri 286	. . . . . . 7 |- ( ( Lim z -> x e. z ) <-> ( z e. On -> ( -. x e. z -> ( z e. On /\ -. Lim z ) ) ) )
24	11, 23	syl6rbbr 279	. . . . . 6 |- ( x e. On -> ( ( Lim z -> x e. z ) <-> ( z e. On -> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) ) )
25		impexp 462	. . . . . . 7 |- ( ( ( z e. On /\ z e. suc x ) -> z e. { y e. On | -. Lim y } ) <-> ( z e. On -> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) )
26		simpr 477	. . . . . . . . 9 |- ( ( z e. On /\ z e. suc x ) -> z e. suc x )
27		suceloni 7013	. . . . . . . . . . 11 |- ( x e. On -> suc x e. On )
28		onelon 5748	. . . . . . . . . . . 12 |- ( ( suc x e. On /\ z e. suc x ) -> z e. On )
29	28	ex 450	. . . . . . . . . . 11 |- ( suc x e. On -> ( z e. suc x -> z e. On ) )
30	27, 29	syl 17	. . . . . . . . . 10 |- ( x e. On -> ( z e. suc x -> z e. On ) )
31	30	ancrd 577	. . . . . . . . 9 |- ( x e. On -> ( z e. suc x -> ( z e. On /\ z e. suc x ) ) )
32	26, 31	impbid2 216	. . . . . . . 8 |- ( x e. On -> ( ( z e. On /\ z e. suc x ) <-> z e. suc x ) )
33	32	imbi1d 331	. . . . . . 7 |- ( x e. On -> ( ( ( z e. On /\ z e. suc x ) -> z e. { y e. On | -. Lim y } ) <-> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) )
34	25, 33	syl5bbr 274	. . . . . 6 |- ( x e. On -> ( ( z e. On -> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) <-> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) )
35	24, 34	bitrd 268	. . . . 5 |- ( x e. On -> ( ( Lim z -> x e. z ) <-> ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) )
36	35	albidv 1849	. . . 4 |- ( x e. On -> ( A. z ( Lim z -> x e. z ) <-> A. z ( z e. suc x -> z e. { y e. On | -. Lim y } ) ) )
37		dfss2 3591	. . . 4 |- ( suc x C_ { y e. On | -. Lim y } <-> A. z ( z e. suc x -> z e. { y e. On | -. Lim y } ) )
38	36, 37	syl6bbr 278	. . 3 |- ( x e. On -> ( A. z ( Lim z -> x e. z ) <-> suc x C_ { y e. On | -. Lim y } ) )
39	38	rabbiia 3185	. 2 |- { x e. On | A. z ( Lim z -> x e. z ) } = { x e. On | suc x C_ { y e. On | -. Lim y } }
40	1, 39	eqtri 2644	1 |- om = { x e. On | suc x C_ { y e. On | -. Lim y } }

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   Oncon0 5723   Lim wlim 5724   suc csuc 5725   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  omsson  7069