Theorem rankr1id 5808

Description: The rank of the hierarchy of an ordinal number is itself.

Assertion

Ref	Expression
rankr1id	|- (A e. On <-> (rank` (R1` A)) = A)

Proof of Theorem rankr1id

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . 5 |- (x = A -> (R1` x) = (R1` A))
2	1	fveq2d 4685	. . . 4 |- (x = A -> (rank` (R1` x)) = (rank` (R1` A)))
3		id 73	. . . 4 |- (x = A -> x = A)
4	2, 3	eqeq12d 1899	. . 3 |- (x = A -> ((rank` (R1` x)) = x <-> (rank` (R1` A)) = A))
5		r1ord3 5768	. . . . . . . 8 |- ((x e. On /\ y e. On) -> (x C_ y -> (R1` x) C_ (R1` y)))
6	5	ss2rabdv 2687	. . . . . . 7 |- (x e. On -> {y e. On | x C_ y} C_ {y e. On | (R1` x) C_ (R1` y)})
7		intss 3239	. . . . . . 7 |- ({y e. On | x C_ y} C_ {y e. On | (R1` x) C_ (R1` y)} -> |^|{y e. On | (R1` x) C_ (R1` y)} C_ |^|{y e. On | x C_ y})
8	6, 7	syl 12	. . . . . 6 |- (x e. On -> |^|{y e. On | (R1` x) C_ (R1` y)} C_ |^|{y e. On | x C_ y})
9		intmin 3237	. . . . . 6 |- (x e. On -> |^|{y e. On | x C_ y} = x)
10	8, 9	sseqtrd 2653	. . . . 5 |- (x e. On -> |^|{y e. On | (R1` x) C_ (R1` y)} C_ x)
11		fvex 4689	. . . . . 6 |- (R1` x) e. _V
12		rankval2 5781	. . . . . 6 |- ((R1` x) e. _V -> (rank` (R1` x)) = |^|{y e. On | (R1` x) C_ (R1` y)})
13	11, 12	ax-mp 7	. . . . 5 |- (rank` (R1` x)) = |^|{y e. On | (R1` x) C_ (R1` y)}
14	10, 13	syl5ss 2661	. . . 4 |- (x e. On -> (rank` (R1` x)) C_ x)
15		rankonid 5806	. . . . 5 |- (x e. On <-> (rank` x) = x)
16		visset 2295	. . . . . . . . 9 |- x e. _V
17		r1rankid 5805	. . . . . . . . 9 |- (x e. _V -> x C_ (R1` (rank` x)))
18	16, 17	ax-mp 7	. . . . . . . 8 |- x C_ (R1` (rank` x))
19		fveq2 4681	. . . . . . . . 9 |- ((rank` x) = x -> (R1` (rank` x)) = (R1` x))
20	19	sseq2d 2645	. . . . . . . 8 |- ((rank` x) = x -> (x C_ (R1` (rank` x)) <-> x C_ (R1` x)))
21	18, 20	mpbii 210	. . . . . . 7 |- ((rank` x) = x -> x C_ (R1` x))
22	11	rankss 5799	. . . . . . 7 |- (x C_ (R1` x) -> (rank` x) C_ (rank` (R1` x)))
23	21, 22	syl 12	. . . . . 6 |- ((rank` x) = x -> (rank` x) C_ (rank` (R1` x)))
24		sseq1 2637	. . . . . 6 |- ((rank` x) = x -> ((rank` x) C_ (rank` (R1` x)) <-> x C_ (rank` (R1` x))))
25	23, 24	mpbid 212	. . . . 5 |- ((rank` x) = x -> x C_ (rank` (R1` x)))
26	15, 25	sylbi 216	. . . 4 |- (x e. On -> x C_ (rank` (R1` x)))
27	14, 26	eqssd 2633	. . 3 |- (x e. On -> (rank` (R1` x)) = x)
28	4, 27	vtoclga 2352	. 2 |- (A e. On -> (rank` (R1` A)) = A)
29		rankon 5782	. . 3 |- (rank` (R1` A)) e. On
30		eleq1 1957	. . 3 |- ((rank` (R1` A)) = A -> ((rank` (R1` A)) e. On <-> A e. On))
31	29, 30	mpbii 210	. 2 |- ((rank` (R1` A)) = A -> A e. On)
32	28, 31	impbii 174	1 |- (A e. On <-> (rank` (R1` A)) = A)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   C_ wss 2593  |^|cint 3214  Oncon0 3657  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankuni 5809  rankr1b 5810  rankelun 5818

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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