Theorem rankval3 3525

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79.

Hypothesis

Ref	Expression
rankval3.1	|- A e. V

Assertion

Ref	Expression
rankval3	|- (rank` A) = |^|{x e. On | A.y e. A (rank` y) e. x}

Distinct variable group(s):   x,y,A

Proof of Theorem rankval3

Step	Hyp	Ref	Expression
1		rankval3.1	. . . 4 |- A e. V
2	1	rankval 3512	. . 3 |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}
3		visset 1350	. . . . . . . . 9 |- y e. V
4	3	rankid 3516	. . . . . . . 8 |- y e. (R1` suc (rank` y))
5		eloni 2209	. . . . . . . . . . 11 |- (x e. On -> Ord x)
6		ordsucss 2320	. . . . . . . . . . 11 |- (Ord x -> ((rank` y) e. x -> suc (rank` y) (_ x))
7	5, 6	syl 12	. . . . . . . . . 10 |- (x e. On -> ((rank` y) e. x -> suc (rank` y) (_ x))
8		rankon 3515	. . . . . . . . . . . 12 |- (rank` y) e. On
9	8	onsuc 2353	. . . . . . . . . . 11 |- suc (rank` y) e. On
10		r1ord3 3501	. . . . . . . . . . 11 |- ((suc (rank` y) e. On /\ x e. On) -> (suc (rank` y) (_ x -> (R1` suc (rank` y)) (_ (R1` x)))
11	9, 10	mpan 518	. . . . . . . . . 10 |- (x e. On -> (suc (rank` y) (_ x -> (R1` suc (rank` y)) (_ (R1` x)))
12	7, 11	syld 27	. . . . . . . . 9 |- (x e. On -> ((rank` y) e. x -> (R1` suc (rank` y)) (_ (R1` x)))
13		ssel 1502	. . . . . . . . 9 |- ((R1` suc (rank` y)) (_ (R1` x) -> (y e. (R1` suc (rank` y)) -> y e. (R1` x)))
14	12, 13	syl6 23	. . . . . . . 8 |- (x e. On -> ((rank` y) e. x -> (y e. (R1` suc (rank` y)) -> y e. (R1` x))))
15	4, 14	mpii 45	. . . . . . 7 |- (x e. On -> ((rank` y) e. x -> y e. (R1` x)))
16	15	r19.20sdv 1257	. . . . . 6 |- (x e. On -> (A.y e. A (rank` y) e. x -> A.y e. A y e. (R1` x)))
17		r1suc 3496	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
18	17	eleq2d 1156	. . . . . . 7 |- (x e. On -> (A e. (R1` suc x) <-> A e. P~(R1` x)))
19	1	elpw 1801	. . . . . . . 8 |- (A e. P~(R1` x) <-> A (_ (R1` x))
20		dfss3 1498	. . . . . . . 8 |- (A (_ (R1` x) <-> A.y e. A y e. (R1` x))
21	19, 20	bitr 151	. . . . . . 7 |- (A e. P~(R1` x) <-> A.y e. A y e. (R1` x))
22	18, 21	syl6bb 414	. . . . . 6 |- (x e. On -> (A e. (R1` suc x) <-> A.y e. A y e. (R1` x)))
23	16, 22	sylibrd 179	. . . . 5 |- (x e. On -> (A.y e. A (rank` y) e. x -> A e. (R1` suc x)))
24	23	ss2rabi 1554	. . . 4 |- {x e. On | A.y e. A (rank` y) e. x} (_ {x e. On | A e. (R1` suc x)}
25		intss 1983	. . . 4 |- ({x e. On | A.y e. A (rank` y) e. x} (_ {x e. On | A e. (R1` suc x)} -> |^|{x e. On | A e. (R1` suc x)} (_ |^|{x e. On | A.y e. A (rank` y) e. x})
26	24, 25	ax-mp 6	. . 3 |- |^|{x e. On | A e. (R1` suc x)} (_ |^|{x e. On | A.y e. A (rank` y) e. x}
27	2, 26	eqsstr 1530	. 2 |- (rank` A) (_ |^|{x e. On | A.y e. A (rank` y) e. x}
28		rankon 3515	. . 3 |- (rank` A) e. On
29	1	rankel 3524	. . . 4 |- (y e. A -> (rank` y) e. (rank` A))
30	29	rgen 1247	. . 3 |- A.y e. A (rank` y) e. (rank` A)
31		eleq2 1150	. . . . 5 |- (x = (rank` A) -> ((rank` y) e. x <-> (rank` y) e. (rank` A)))
32	31	biraldv 1219	. . . 4 |- (x = (rank` A) -> (A.y e. A (rank` y) e. x <-> A.y e. A (rank` y) e. (rank` A)))
33	32	onintss 2266	. . 3 |- ((rank` A) e. On -> (A.y e. A (rank` y) e. (rank` A) -> |^|{x e. On | A.y e. A (rank` y) e. x} (_ (rank` A)))
34	28, 30, 33	mp2 43	. 2 |- |^|{x e. On | A.y e. A (rank` y) e. x} (_ (rank` A)
35	27, 34	eqssi 1517	1 |- (rank` A) = |^|{x e. On | A.y e. A (rank` y) e. x}

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204  Vcvv 1348   (_ wss 1487  P~cpw 1798  |^|cint 1965  Ord word 2198  Oncon0 2199  suc csuc 2201  ` cfv 2422  R1cr1 3485  rankcrnk 3486

This theorem is referenced by:  ranksn 3532  rankuni 3533  rankun 3535  rankonid 3538

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

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