Theorem rankval3 4661

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 4648	. . 3 |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}
3		visset 1809	. . . . . . . . 9 |- y e. V
4	3	rankid 4652	. . . . . . . 8 |- y e. (R1` suc (rank` y))
5		eloni 2953	. . . . . . . . . . 11 |- (x e. On -> Ord x)
6		ordsucss 3064	. . . . . . . . . . 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 4651	. . . . . . . . . . . 12 |- (rank` y) e. On
9	8	onsuc 3100	. . . . . . . . . . 11 |- suc (rank` y) e. On
10		r1ord3 4637	. . . . . . . . . . 11 |- ((suc (rank` y) e. On /\ x e. On) -> (suc (rank` y) (_ x -> (R1` suc (rank` y)) (_ (R1` x)))
11	9, 10	mpan 694	. . . . . . . . . 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 2059	. . . . . . . . 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 1707	. . . . . 6 |- (x e. On -> (A.y e. A (rank` y) e. x -> A.y e. A y e. (R1` x)))
17		r1suc 4632	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
18	17	eleq2d 1538	. . . . . . 7 |- (x e. On -> (A e. (R1` suc x) <-> A e. P~(R1` x)))
19	1	elpw 2400	. . . . . . . 8 |- (A e. P~(R1` x) <-> A (_ (R1` x))
20		dfss3 2055	. . . . . . . 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 535	. . . . . 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 2124	. . . 4 |- {x e. On | A.y e. A (rank` y) e. x} (_ {x e. On | A e. (R1` suc x)}
25		intss 2549	. . . 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 2087	. 2 |- (rank` A) (_ |^|{x e. On | A.y e. A (rank` y) e. x}
28		rankon 4651	. . 3 |- (rank` A) e. On
29	1	rankel 4660	. . . 4 |- (y e. A -> (rank` y) e. (rank` A))
30	29	rgen 1695	. . 3 |- A.y e. A (rank` y) e. (rank` A)
31		eleq2 1532	. . . . 5 |- (x = (rank` A) -> ((rank` y) e. x <-> (rank` y) e. (rank` A)))
32	31	ralbidv 1660	. . . 4 |- (x = (rank` A) -> (A.y e. A (rank` y) e. x <-> A.y e. A (rank` y) e. (rank` A)))
33	32	onintss 3006	. . 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 2074	1 |- (rank` A) = |^|{x e. On | A.y e. A (rank` y) e. x}

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  A.wral 1642  {crab 1645  Vcvv 1807   (_ wss 2043  P~cpw 2397  |^|cint 2528  Ord word 2942  Oncon0 2943  suc csuc 2945  ` cfv 3177  R1cr1 4621  rankcrnk 4622

This theorem is referenced by:  ranksn 4669  rankuni2 4670  rankun 4671  rankonid 4675

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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