Theorem rankr1id 4669

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 3709	. . . . 5 |- (x = A -> (R1` x) = (R1` A))
2	1	fveq2d 3713	. . . 4 |- (x = A -> (rank` (R1` x)) = (rank` (R1` A)))
3		id 59	. . . 4 |- (x = A -> x = A)
4	2, 3	eqeq12d 1481	. . 3 |- (x = A -> ((rank` (R1` x)) = x <-> (rank` (R1` A)) = A))
5		r1ord3 4629	. . . . . . . 8 |- ((x e. On /\ y e. On) -> (x (_ y -> (R1` x) (_ (R1` y)))
6	5	ss2rabdv 2117	. . . . . . 7 |- (x e. On -> {y e. On | x (_ y} (_ {y e. On | (R1` x) (_ (R1` y)})
7		intss 2544	. . . . . . 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 2543	. . . . . 6 |- (x e. On -> |^|{y e. On | x (_ y} = x)
10	8, 9	sseqtrd 2087	. . . . 5 |- (x e. On -> |^|{y e. On | (R1` x) (_ (R1` y)} (_ x)
11		fvex 3717	. . . . . 6 |- (R1` x) e. V
12		rankval2 4642	. . . . . 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 2095	. . . 4 |- (x e. On -> (rank` (R1` x)) (_ x)
15		rankonid 4667	. . . . 5 |- (x e. On <-> (rank` x) = x)
16		visset 1804	. . . . . . . . 9 |- x e. V
17		r1rankid 4666	. . . . . . . . 9 |- (x e. V -> x (_ (R1` (rank` x)))
18	16, 17	ax-mp 7	. . . . . . . 8 |- x (_ (R1` (rank` x))
19		fveq2 3709	. . . . . . . . 9 |- ((rank` x) = x -> (R1` (rank` x)) = (R1` x))
20	19	sseq2d 2079	. . . . . . . 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 4660	. . . . . . 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 2072	. . . . . 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 2069	. . 3 |- (x e. On -> (rank` (R1` x)) = x)
28	4, 27	vtoclga 1843	. 2 |- (A e. On -> (rank` (R1` A)) = A)
29		rankon 4643	. . 3 |- (rank` (R1` A)) e. On
30		eleq1 1526	. . 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 953   e. wcel 955  {crab 1640  Vcvv 1802   (_ wss 2037  |^|cint 2523  Oncon0 2938  ` cfv 3172  R1cr1 4613  rankcrnk 4614

This theorem is referenced by:  rankuni 4670  rankr1b 4671  rankelun 4679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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