Theorem rankval3 4681

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:   x,y,A

Proof of Theorem rankval3

Step	Hyp	Ref	Expression
1		rankval3.1	. . . 4 |- A e. V
2	1	rankval 4668	. . 3 |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}
3		visset 1813	. . . . . . . . 9 |- y e. V
4	3	rankid 4672	. . . . . . . 8 |- y e. (R1` suc (rank` y))
5		eloni 2958	. . . . . . . . . . 11 |- (x e. On -> Ord x)
6		ordsucss 3069	. . . . . . . . . . 11 |- (Ord x -> ((rank` y) e. x -> suc (rank` y) (_ x))
7	5, 6	syl 10	. . . . . . . . . 10 |- (x e. On -> ((rank` y) e. x -> suc (rank` y) (_ x))
8		rankon 4671	. . . . . . . . . . . 12 |- (rank` y) e. On
9	8	onsuc 3105	. . . . . . . . . . 11 |- suc (rank` y) e. On
10		r1ord3 4657	. . . . . . . . . . 11 |- ((suc (rank` y) e. On /\ x e. On) -> (suc (rank` y) (_ x -> (R1` suc (rank` y)) (_ (R1` x)))
11	9, 10	mpan 695	. . . . . . . . . 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 2063	. . . . . . . . 9 |- ((R1` suc (rank` y)) (_ (R1` x) -> (y e. (R1` suc (rank` y)) -> y e. (R1` x)))
14	12, 13	syl6 22	. . . . . . . 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 1710	. . . . . 6 |- (x e. On -> (A.y e. A (rank` y) e. x -> A.y e. A y e. (R1` x)))
17		r1suc 4652	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
18	17	eleq2d 1541	. . . . . . 7 |- (x e. On -> (A e. (R1` suc x) <-> A e. P~(R1` x)))
19	1	elpw 2404	. . . . . . . 8 |- (A e. P~(R1` x) <-> A (_ (R1` x))
20		dfss3 2059	. . . . . . . 8 |- (A (_ (R1` x) <-> A.y e. A y e. (R1` x))
21	19, 20	bitr 173	. . . . . . 7 |- (A e. P~(R1` x) <-> A.y e. A y e. (R1` x))
22	18, 21	syl6bb 536	. . . . . 6 |- (x e. On -> (A e. (R1` suc x) <-> A.y e. A y e. (R1` x)))
23	16, 22	sylibrd 204	. . . . 5 |- (x e. On -> (A.y e. A (rank` y) e. x -> A e. (R1` suc x)))
24	23	ss2rabi 2128	. . . 4 |- {x e. On | A.y e. A (rank` y) e. x} (_ {x e. On | A e. (R1` suc x)}
25		intss 2554	. . . 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 7	. . 3 |- |^|{x e. On | A e. (R1` suc x)} (_ |^|{x e. On | A.y e. A (rank` y) e. x}
27	2, 26	eqsstr 2091	. 2 |- (rank` A) (_ |^|{x e. On | A.y e. A (rank` y) e. x}
28		rankon 4671	. . 3 |- (rank` A) e. On
29	1	rankel 4680	. . . 4 |- (y e. A -> (rank` y) e. (rank` A))
30	29	rgen 1698	. . 3 |- A.y e. A (rank` y) e. (rank` A)
31		eleq2 1535	. . . . 5 |- (x = (rank` A) -> ((rank` y) e. x <-> (rank` y) e. (rank` A)))
32	31	ralbidv 1663	. . . 4 |- (x = (rank` A) -> (A.y e. A (rank` y) e. x <-> A.y e. A (rank` y) e. (rank` A)))
33	32	onintss 3011	. . 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 2078	1 |- (rank` A) = |^|{x e. On | A.y e. A (rank` y) e. x}

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811   (_ wss 2047  P~cpw 2401  |^|cint 2533  Ord word 2947  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  ranksn 4689  rankuni2 4690  rankun 4691  rankonid 4695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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