Theorem r1wunlim 9559

Description: The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
r1wunlim	|- ( A e. V -> ( ( R1 ` A ) e. WUni <-> Lim A ) )

Proof of Theorem r1wunlim

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> ( R1 ` A ) e. WUni )
2	1	wun0 9540	. . . . . 6 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> (/) e. ( R1 ` A ) )
3		elfvdm 6220	. . . . . 6 |- ( (/) e. ( R1 ` A ) -> A e. dom R1 )
4	2, 3	syl 17	. . . . 5 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> A e. dom R1 )
5		r1fnon 8630	. . . . . 6 |- R1 Fn On
6		fndm 5990	. . . . . 6 |- ( R1 Fn On -> dom R1 = On )
7	5, 6	ax-mp 5	. . . . 5 |- dom R1 = On
8	4, 7	syl6eleq 2711	. . . 4 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> A e. On )
9		eloni 5733	. . . 4 |- ( A e. On -> Ord A )
10	8, 9	syl 17	. . 3 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> Ord A )
11		n0i 3920	. . . . . 6 |- ( (/) e. ( R1 ` A ) -> -. ( R1 ` A ) = (/) )
12	2, 11	syl 17	. . . . 5 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> -. ( R1 ` A ) = (/) )
13		fveq2 6191	. . . . . 6 |- ( A = (/) -> ( R1 ` A ) = ( R1 ` (/) ) )
14		r10 8631	. . . . . 6 |- ( R1 ` (/) ) = (/)
15	13, 14	syl6eq 2672	. . . . 5 |- ( A = (/) -> ( R1 ` A ) = (/) )
16	12, 15	nsyl 135	. . . 4 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> -. A = (/) )
17		suceloni 7013	. . . . . . . 8 |- ( A e. On -> suc A e. On )
18	8, 17	syl 17	. . . . . . 7 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> suc A e. On )
19		sucidg 5803	. . . . . . . 8 |- ( A e. On -> A e. suc A )
20	8, 19	syl 17	. . . . . . 7 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> A e. suc A )
21		r1ord 8643	. . . . . . 7 |- ( suc A e. On -> ( A e. suc A -> ( R1 ` A ) e. ( R1 ` suc A ) ) )
22	18, 20, 21	sylc 65	. . . . . 6 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> ( R1 ` A ) e. ( R1 ` suc A ) )
23		r1elwf 8659	. . . . . 6 |- ( ( R1 ` A ) e. ( R1 ` suc A ) -> ( R1 ` A ) e. U. ( R1 " On ) )
24		wfelirr 8688	. . . . . 6 |- ( ( R1 ` A ) e. U. ( R1 " On ) -> -. ( R1 ` A ) e. ( R1 ` A ) )
25	22, 23, 24	3syl 18	. . . . 5 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> -. ( R1 ` A ) e. ( R1 ` A ) )
26		simprr 796	. . . . . . . . 9 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> A = suc x )
27	26	fveq2d 6195	. . . . . . . 8 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ( R1 ` A ) = ( R1 ` suc x ) )
28		r1suc 8633	. . . . . . . . 9 |- ( x e. On -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
29	28	ad2antrl 764	. . . . . . . 8 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
30	27, 29	eqtrd 2656	. . . . . . 7 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
31		simplr 792	. . . . . . . 8 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ( R1 ` A ) e. WUni )
32	8	adantr 481	. . . . . . . . 9 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> A e. On )
33		sucidg 5803	. . . . . . . . . . 11 |- ( x e. On -> x e. suc x )
34	33	ad2antrl 764	. . . . . . . . . 10 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> x e. suc x )
35	34, 26	eleqtrrd 2704	. . . . . . . . 9 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> x e. A )
36		r1ord 8643	. . . . . . . . 9 |- ( A e. On -> ( x e. A -> ( R1 ` x ) e. ( R1 ` A ) ) )
37	32, 35, 36	sylc 65	. . . . . . . 8 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ( R1 ` x ) e. ( R1 ` A ) )
38	31, 37	wunpw 9529	. . . . . . 7 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ~P ( R1 ` x ) e. ( R1 ` A ) )
39	30, 38	eqeltrd 2701	. . . . . 6 |- ( ( ( A e. V /\ ( R1 ` A ) e. WUni ) /\ ( x e. On /\ A = suc x ) ) -> ( R1 ` A ) e. ( R1 ` A ) )
40	39	rexlimdvaa 3032	. . . . 5 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> ( E. x e. On A = suc x -> ( R1 ` A ) e. ( R1 ` A ) ) )
41	25, 40	mtod 189	. . . 4 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> -. E. x e. On A = suc x )
42		ioran 511	. . . 4 |- ( -. ( A = (/) \/ E. x e. On A = suc x ) <-> ( -. A = (/) /\ -. E. x e. On A = suc x ) )
43	16, 41, 42	sylanbrc 698	. . 3 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> -. ( A = (/) \/ E. x e. On A = suc x ) )
44		dflim3 7047	. . 3 |- ( Lim A <-> ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) )
45	10, 43, 44	sylanbrc 698	. 2 |- ( ( A e. V /\ ( R1 ` A ) e. WUni ) -> Lim A )
46		r1limwun 9558	. 2 |- ( ( A e. V /\ Lim A ) -> ( R1 ` A ) e. WUni )
47	45, 46	impbida 877	1 |- ( A e. V -> ( ( R1 ` A ) e. WUni <-> Lim A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   dom cdm 5114   "cima 5117   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fn wfn 5883   ` cfv 5888   R1cr1 8625  WUnicwun 9522

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628  df-wun 9524

This theorem is referenced by: (None)