Theorem rankr1id 3539

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 2832	. . . . 5 |- (x = A -> (R1` x) = (R1` A))
2	1	fveq2d 2836	. . . 4 |- (x = A -> (rank` (R1` x)) = (rank` (R1` A)))
3		id 9	. . . 4 |- (x = A -> x = A)
4	2, 3	cleq12d 1115	. . 3 |- (x = A -> ((rank` (R1` x)) = x <-> (rank` (R1` A)) = A))
5		r1ord3 3501	. . . . . . . . . 10 |- ((x e. On /\ y e. On) -> (x (_ y -> (R1` x) (_ (R1` y)))
6	5	exp 291	. . . . . . . . 9 |- (x e. On -> (y e. On -> (x (_ y -> (R1` x) (_ (R1` y))))
7	6	r19.21aiv 1259	. . . . . . . 8 |- (x e. On -> A.y e. On (x (_ y -> (R1` x) (_ (R1` y)))
8		ss2rab 1553	. . . . . . . 8 |- ({y e. On | x (_ y} (_ {y e. On | (R1` x) (_ (R1` y)} <-> A.y e. On (x (_ y -> (R1` x) (_ (R1` y)))
9	7, 8	sylibr 175	. . . . . . 7 |- (x e. On -> {y e. On | x (_ y} (_ {y e. On | (R1` x) (_ (R1` y)})
10		intss 1983	. . . . . . 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})
11	9, 10	syl 12	. . . . . 6 |- (x e. On -> |^|{y e. On | (R1` x) (_ (R1` y)} (_ |^|{y e. On | x (_ y})
12		intmin 1982	. . . . . 6 |- (x e. On -> x = |^|{y e. On | x (_ y})
13	11, 12	sseqtr4d 1537	. . . . 5 |- (x e. On -> |^|{y e. On | (R1` x) (_ (R1` y)} (_ x)
14		fvex 2838	. . . . . 6 |- (R1` x) e. V
15		rankval2 3514	. . . . . 6 |- ((R1` x) e. V -> (rank` (R1` x)) = |^|{y e. On | (R1` x) (_ (R1` y)})
16	14, 15	ax-mp 6	. . . . 5 |- (rank` (R1` x)) = |^|{y e. On | (R1` x) (_ (R1` y)}
17	13, 16	syl5ss 1544	. . . 4 |- (x e. On -> (rank` (R1` x)) (_ x)
18		rankonid 3538	. . . . 5 |- (x e. On <-> (rank` x) = x)
19		visset 1350	. . . . . . . . 9 |- x e. V
20		r1rankid 3537	. . . . . . . . 9 |- (x e. V -> x (_ (R1` (rank` x)))
21	19, 20	ax-mp 6	. . . . . . . 8 |- x (_ (R1` (rank` x))
22		fveq2 2832	. . . . . . . . 9 |- ((rank` x) = x -> (R1` (rank` x)) = (R1` x))
23	22	sseq2d 1528	. . . . . . . 8 |- ((rank` x) = x -> (x (_ (R1` (rank` x)) <-> x (_ (R1` x)))
24	21, 23	mpbii 168	. . . . . . 7 |- ((rank` x) = x -> x (_ (R1` x))
25	14	rankss 3531	. . . . . . 7 |- (x (_ (R1` x) -> (rank` x) (_ (rank` (R1` x)))
26	24, 25	syl 12	. . . . . 6 |- ((rank` x) = x -> (rank` x) (_ (rank` (R1` x)))
27		sseq1 1521	. . . . . 6 |- ((rank` x) = x -> ((rank` x) (_ (rank` (R1` x)) <-> x (_ (rank` (R1` x))))
28	26, 27	mpbid 170	. . . . 5 |- ((rank` x) = x -> x (_ (rank` (R1` x)))
29	18, 28	sylbi 174	. . . 4 |- (x e. On -> x (_ (rank` (R1` x)))
30	17, 29	eqssd 1518	. . 3 |- (x e. On -> (rank` (R1` x)) = x)
31	4, 30	vtoclga 1387	. 2 |- (A e. On -> (rank` (R1` A)) = A)
32		rankon 3515	. . 3 |- (rank` (R1` A)) e. On
33		eleq1 1149	. . 3 |- ((rank` (R1` A)) = A -> ((rank` (R1` A)) e. On <-> A e. On))
34	32, 33	mpbii 168	. 2 |- ((rank` (R1` A)) = A -> A e. On)
35	31, 34	impbi 139	1 |- (A e. On <-> (rank` (R1` A)) = A)

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204  Vcvv 1348   (_ wss 1487  |^|cint 1965  Oncon0 2199  ` cfv 2422  R1cr1 3485  rankcrnk 3486

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

metamath.org