Theorem rankval4 4702

Description: The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204.

Hypothesis

Ref	Expression
rankr1b.1	|- A e. V

Assertion

Ref	Expression
rankval4	|- (rank` A) = U_x e. A suc (rank` x)

Distinct variable group:   x,A

Proof of Theorem rankval4

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . . 6 |- (y e. A -> A.x y e. A)
2		ax-17 971	. . . . . . 7 |- (y e. R1 -> A.x y e. R1)
3		hbiu1 2584	. . . . . . 7 |- (y e. U_x e. A suc (rank` x) -> A.x y e. U_x e. A suc (rank` x))
4	2, 3	hbfv 3729	. . . . . 6 |- (y e. (R1` U_x e. A suc (rank` x)) -> A.x y e. (R1` U_x e. A suc (rank` x)))
5	1, 4	dfss2f 2060	. . . . 5 |- (A (_ (R1` U_x e. A suc (rank` x)) <-> A.x(x e. A -> x e. (R1` U_x e. A suc (rank` x))))
6		visset 1813	. . . . . . 7 |- x e. V
7	6	rankid 4672	. . . . . 6 |- x e. (R1` suc (rank` x))
8		ssiun2 2593	. . . . . . . 8 |- (x e. A -> suc (rank` x) (_ U_x e. A suc (rank` x))
9		rankon 4671	. . . . . . . . . 10 |- (rank` x) e. On
10	9	onsuc 3105	. . . . . . . . 9 |- suc (rank` x) e. On
11		rankr1b.1	. . . . . . . . . . 11 |- A e. V
12		fvex 3732	. . . . . . . . . . . 12 |- (rank` x) e. V
13	12	sucex 3050	. . . . . . . . . . 11 |- suc (rank` x) e. V
14	11, 13	iunon 3909	. . . . . . . . . 10 |- (A.x e. A suc (rank` x) e. On -> U_x e. A suc (rank` x) e. On)
15	10	a1i 8	. . . . . . . . . 10 |- (x e. A -> suc (rank` x) e. On)
16	14, 15	mprg 1700	. . . . . . . . 9 |- U_x e. A suc (rank` x) e. On
17		r1ord3 4657	. . . . . . . . 9 |- ((suc (rank` x) e. On /\ U_x e. A suc (rank` x) e. On) -> (suc (rank` x) (_ U_x e. A suc (rank` x) -> (R1` suc (rank` x)) (_ (R1` U_x e. A suc (rank` x))))
18	10, 16, 17	mp2an 697	. . . . . . . 8 |- (suc (rank` x) (_ U_x e. A suc (rank` x) -> (R1` suc (rank` x)) (_ (R1` U_x e. A suc (rank` x)))
19	8, 18	syl 10	. . . . . . 7 |- (x e. A -> (R1` suc (rank` x)) (_ (R1` U_x e. A suc (rank` x)))
20	19	sseld 2067	. . . . . 6 |- (x e. A -> (x e. (R1` suc (rank` x)) -> x e. (R1` U_x e. A suc (rank` x))))
21	7, 20	mpi 44	. . . . 5 |- (x e. A -> x e. (R1` U_x e. A suc (rank` x)))
22	5, 21	mpgbir 988	. . . 4 |- A (_ (R1` U_x e. A suc (rank` x))
23		fvex 3732	. . . . 5 |- (R1` U_x e. A suc (rank` x)) e. V
24	23	rankss 4688	. . . 4 |- (A (_ (R1` U_x e. A suc (rank` x)) -> (rank` A) (_ (rank` (R1` U_x e. A suc (rank` x))))
25	22, 24	ax-mp 7	. . 3 |- (rank` A) (_ (rank` (R1` U_x e. A suc (rank` x)))
26		r1ord3 4657	. . . . . . 7 |- ((U_x e. A suc (rank` x) e. On /\ y e. On) -> (U_x e. A suc (rank` x) (_ y -> (R1` U_x e. A suc (rank` x)) (_ (R1` y)))
27	16, 26	mpan 695	. . . . . 6 |- (y e. On -> (U_x e. A suc (rank` x) (_ y -> (R1` U_x e. A suc (rank` x)) (_ (R1` y)))
28	27	ss2rabi 2128	. . . . 5 |- {y e. On | U_x e. A suc (rank` x) (_ y} (_ {y e. On | (R1` U_x e. A suc (rank` x)) (_ (R1` y)}
29		intss 2554	. . . . 5 |- ({y e. On | U_x e. A suc (rank` x) (_ y} (_ {y e. On | (R1` U_x e. A suc (rank` x)) (_ (R1` y)} -> |^|{y e. On | (R1` U_x e. A suc (rank` x)) (_ (R1` y)} (_ |^|{y e. On | U_x e. A suc (rank` x) (_ y})
30	28, 29	ax-mp 7	. . . 4 |- |^|{y e. On | (R1` U_x e. A suc (rank` x)) (_ (R1` y)} (_ |^|{y e. On | U_x e. A suc (rank` x) (_ y}
31		rankval2 4670	. . . . 5 |- ((R1` U_x e. A suc (rank` x)) e. V -> (rank` (R1` U_x e. A suc (rank` x))) = |^|{y e. On | (R1` U_x e. A suc (rank` x)) (_ (R1` y)})
32	23, 31	ax-mp 7	. . . 4 |- (rank` (R1` U_x e. A suc (rank` x))) = |^|{y e. On | (R1` U_x e. A suc (rank` x)) (_ (R1` y)}
33		intmin 2553	. . . . . 6 |- (U_x e. A suc (rank` x) e. On -> |^|{y e. On | U_x e. A suc (rank` x) (_ y} = U_x e. A suc (rank` x))
34	16, 33	ax-mp 7	. . . . 5 |- |^|{y e. On | U_x e. A suc (rank` x) (_ y} = U_x e. A suc (rank` x)
35	34	eqcomi 1479	. . . 4 |- U_x e. A suc (rank` x) = |^|{y e. On | U_x e. A suc (rank` x) (_ y}
36	30, 32, 35	3sstr4 2100	. . 3 |- (rank` (R1` U_x e. A suc (rank` x))) (_ U_x e. A suc (rank` x)
37	25, 36	sstri 2073	. 2 |- (rank` A) (_ U_x e. A suc (rank` x)
38		iunss 2591	. . 3 |- (U_x e. A suc (rank` x) (_ (rank` A) <-> A.x e. A suc (rank` x) (_ (rank` A))
39	11	rankel 4680	. . . 4 |- (x e. A -> (rank` x) e. (rank` A))
40		rankon 4671	. . . . 5 |- (rank` A) e. On
41	9, 40	onsucss 3111	. . . 4 |- ((rank` x) e. (rank` A) <-> suc (rank` x) (_ (rank` A))
42	39, 41	sylib 198	. . 3 |- (x e. A -> suc (rank` x) (_ (rank` A))
43	38, 42	mprgbir 1701	. 2 |- U_x e. A suc (rank` x) (_ (rank` A)
44	37, 43	eqssi 2078	1 |- (rank` A) = U_x e. A suc (rank` x)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811   (_ wss 2047  |^|cint 2533  U_ciun 2566  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankbnd 4703  rankc1 4705

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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