Theorem rankr1id 4707

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 3730	. . . . 5 |- (x = A -> (R1` x) = (R1` A))
2	1	fveq2d 3734	. . . 4 |- (x = A -> (rank` (R1` x)) = (rank` (R1` A)))
3		id 59	. . . 4 |- (x = A -> x = A)
4	2, 3	eqeq12d 1492	. . 3 |- (x = A -> ((rank` (R1` x)) = x <-> (rank` (R1` A)) = A))
5		r1ord3 4667	. . . . . . . 8 |- ((x e. On /\ y e. On) -> (x (_ y -> (R1` x) (_ (R1` y)))
6	5	ss2rabdv 2130	. . . . . . 7 |- (x e. On -> {y e. On | x (_ y} (_ {y e. On | (R1` x) (_ (R1` y)})
7		intss 2558	. . . . . . 7 |- ({y e. On | x (_ y} (_ {y e. On | (R1` x) (_ (R1` y)} -> |^|{y e. On | (R1` x) (_ (R1` y)} (_ |^|{y e. On | x (_ y})
8	6, 7	syl 10	. . . . . 6 |- (x e. On -> |^|{y e. On | (R1` x) (_ (R1` y)} (_ |^|{y e. On | x (_ y})
9		intmin 2557	. . . . . 6 |- (x e. On -> |^|{y e. On | x (_ y} = x)
10	8, 9	sseqtrd 2100	. . . . 5 |- (x e. On -> |^|{y e. On | (R1` x) (_ (R1` y)} (_ x)
11		fvex 3738	. . . . . 6 |- (R1` x) e. V
12		rankval2 4680	. . . . . 6 |- ((R1` x) e. V -> (rank` (R1` x)) = |^|{y e. On | (R1` x) (_ (R1` y)})
13	11, 12	ax-mp 7	. . . . 5 |- (rank` (R1` x)) = |^|{y e. On | (R1` x) (_ (R1` y)}
14	10, 13	syl5ss 2108	. . . 4 |- (x e. On -> (rank` (R1` x)) (_ x)
15		rankonid 4705	. . . . 5 |- (x e. On <-> (rank` x) = x)
16		visset 1816	. . . . . . . . 9 |- x e. V
17		r1rankid 4704	. . . . . . . . 9 |- (x e. V -> x (_ (R1` (rank` x)))
18	16, 17	ax-mp 7	. . . . . . . 8 |- x (_ (R1` (rank` x))
19		fveq2 3730	. . . . . . . . 9 |- ((rank` x) = x -> (R1` (rank` x)) = (R1` x))
20	19	sseq2d 2092	. . . . . . . 8 |- ((rank` x) = x -> (x (_ (R1` (rank` x)) <-> x (_ (R1` x)))
21	18, 20	mpbii 193	. . . . . . 7 |- ((rank` x) = x -> x (_ (R1` x))
22	11	rankss 4698	. . . . . . 7 |- (x (_ (R1` x) -> (rank` x) (_ (rank` (R1` x)))
23	21, 22	syl 10	. . . . . 6 |- ((rank` x) = x -> (rank` x) (_ (rank` (R1` x)))
24		sseq1 2085	. . . . . 6 |- ((rank` x) = x -> ((rank` x) (_ (rank` (R1` x)) <-> x (_ (rank` (R1` x))))
25	23, 24	mpbid 195	. . . . 5 |- ((rank` x) = x -> x (_ (rank` (R1` x)))
26	15, 25	sylbi 199	. . . 4 |- (x e. On -> x (_ (rank` (R1` x)))
27	14, 26	eqssd 2082	. . 3 |- (x e. On -> (rank` (R1` x)) = x)
28	4, 27	vtoclga 1855	. 2 |- (A e. On -> (rank` (R1` A)) = A)
29		rankon 4681	. . 3 |- (rank` (R1` A)) e. On
30		eleq1 1537	. . 3 |- ((rank` (R1` A)) = A -> ((rank` (R1` A)) e. On <-> A e. On))
31	29, 30	mpbii 193	. 2 |- ((rank` (R1` A)) = A -> A e. On)
32	28, 31	impbi 157	1 |- (A e. On <-> (rank` (R1` A)) = A)

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  {crab 1651  Vcvv 1814   (_ wss 2050  |^|cint 2537  Oncon0 2954  ` cfv 3188  R1cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankuni 4708  rankr1b 4709  rankelun 4717

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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