Theorem rankval4 4674

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-17 968	. . . . . 6 |- (y e. A -> A.x y e. A)
2		ax-17 968	. . . . . . 7 |- (y e. R1 -> A.x y e. R1)
3		hbiu1 2574	. . . . . . 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 3714	. . . . . 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 2050	. . . . 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 1804	. . . . . . 7 |- x e. V
7	6	rankid 4644	. . . . . 6 |- x e. (R1` suc (rank` x))
8		ssiun2 2583	. . . . . . . 8 |- (x e. A -> suc (rank` x) (_ U_x e. A suc (rank` x))
9		rankon 4643	. . . . . . . . . 10 |- (rank` x) e. On
10	9	onsuc 3095	. . . . . . . . 9 |- suc (rank` x) e. On
11		rankr1b.1	. . . . . . . . . . 11 |- A e. V
12		fvex 3717	. . . . . . . . . . . 12 |- (rank` x) e. V
13	12	sucex 3040	. . . . . . . . . . 11 |- suc (rank` x) e. V
14	11, 13	iunon 3894	. . . . . . . . . 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 1692	. . . . . . . . 9 |- U_x e. A suc (rank` x) e. On
17		r1ord3 4629	. . . . . . . . 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 695	. . . . . . . 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 2057	. . . . . 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 985	. . . 4 |- A (_ (R1` U_x e. A suc (rank` x))
23		fvex 3717	. . . . 5 |- (R1` U_x e. A suc (rank` x)) e. V
24	23	rankss 4660	. . . 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 4629	. . . . . . 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 693	. . . . . 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 2118	. . . . 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 2544	. . . . 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 4642	. . . . 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 2543	. . . . . 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 1471	. . . 4 |- U_x e. A suc (rank` x) = |^|{y e. On | U_x e. A suc (rank` x) (_ y}
36	30, 32, 35	3sstr4 2090	. . 3 |- (rank` (R1` U_x e. A suc (rank` x))) (_ U_x e. A suc (rank` x)
37	25, 36	sstri 2063	. 2 |- (rank` A) (_ U_x e. A suc (rank` x)
38		iunss 2581	. . 3 |- (U_x e. A suc (rank` x) (_ (rank` A) <-> A.x e. A suc (rank` x) (_ (rank` A))
39	11	rankel 4652	. . . 4 |- (x e. A -> (rank` x) e. (rank` A))
40		rankon 4643	. . . . 5 |- (rank` A) e. On
41	9, 40	onsucss 3101	. . . 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 1693	. 2 |- U_x e. A suc (rank` x) (_ (rank` A)
44	37, 43	eqssi 2068	1 |- (rank` A) = U_x e. A suc (rank` x)

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  {crab 1640  Vcvv 1802   (_ wss 2037  |^|cint 2523  U_ciun 2556  Oncon0 2938  suc csuc 2940  ` cfv 3172  R1cr1 4613  rankcrnk 4614

This theorem is referenced by:  rankbnd 4675  rankc1 4677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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