Theorem rankval4 5813

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 1317	. . . . . 6 |- (y e. A -> A.x y e. A)
2		ax-17 1317	. . . . . . 7 |- (y e. R1 -> A.x y e. R1)
3		hbiu1 3281	. . . . . . 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 4686	. . . . . 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 2612	. . . . 5 |- (A C_ (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 2295	. . . . . . 7 |- x e. _V
7	6	rankid 5783	. . . . . 6 |- x e. (R1` suc (rank` x))
8		ssiun2 3295	. . . . . . . 8 |- (x e. A -> suc (rank` x) C_ U_x e. A suc (rank` x))
9		rankon 5782	. . . . . . . . . 10 |- (rank` x) e. On
10	9	onsuci 3919	. . . . . . . . 9 |- suc (rank` x) e. On
11		rankr1b.1	. . . . . . . . . . 11 |- A e. _V
12		fvex 4689	. . . . . . . . . . . 12 |- (rank` x) e. _V
13	12	sucex 3892	. . . . . . . . . . 11 |- suc (rank` x) e. _V
14	11, 13	iunon 5114	. . . . . . . . . 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 2162	. . . . . . . . 9 |- U_x e. A suc (rank` x) e. On
17		r1ord3 5768	. . . . . . . . 9 |- ((suc (rank` x) e. On /\ U_x e. A suc (rank` x) e. On) -> (suc (rank` x) C_ U_x e. A suc (rank` x) -> (R1` suc (rank` x)) C_ (R1` U_x e. A suc (rank` x))))
18	10, 16, 17	mp2an 761	. . . . . . . 8 |- (suc (rank` x) C_ U_x e. A suc (rank` x) -> (R1` suc (rank` x)) C_ (R1` U_x e. A suc (rank` x)))
19	8, 18	syl 12	. . . . . . 7 |- (x e. A -> (R1` suc (rank` x)) C_ (R1` U_x e. A suc (rank` x)))
20	19	sseld 2619	. . . . . 6 |- (x e. A -> (x e. (R1` suc (rank` x)) -> x e. (R1` U_x e. A suc (rank` x))))
21	7, 20	mpi 55	. . . . 5 |- (x e. A -> x e. (R1` U_x e. A suc (rank` x)))
22	5, 21	mpgbir 1334	. . . 4 |- A C_ (R1` U_x e. A suc (rank` x))
23		fvex 4689	. . . . 5 |- (R1` U_x e. A suc (rank` x)) e. _V
24	23	rankss 5799	. . . 4 |- (A C_ (R1` U_x e. A suc (rank` x)) -> (rank` A) C_ (rank` (R1` U_x e. A suc (rank` x))))
25	22, 24	ax-mp 7	. . 3 |- (rank` A) C_ (rank` (R1` U_x e. A suc (rank` x)))
26		r1ord3 5768	. . . . . . 7 |- ((U_x e. A suc (rank` x) e. On /\ y e. On) -> (U_x e. A suc (rank` x) C_ y -> (R1` U_x e. A suc (rank` x)) C_ (R1` y)))
27	16, 26	mpan 759	. . . . . 6 |- (y e. On -> (U_x e. A suc (rank` x) C_ y -> (R1` U_x e. A suc (rank` x)) C_ (R1` y)))
28	27	ss2rabi 2688	. . . . 5 |- {y e. On | U_x e. A suc (rank` x) C_ y} C_ {y e. On | (R1` U_x e. A suc (rank` x)) C_ (R1` y)}
29		intss 3239	. . . . 5 |- ({y e. On | U_x e. A suc (rank` x) C_ y} C_ {y e. On | (R1` U_x e. A suc (rank` x)) C_ (R1` y)} -> |^|{y e. On | (R1` U_x e. A suc (rank` x)) C_ (R1` y)} C_ |^|{y e. On | U_x e. A suc (rank` x) C_ y})
30	28, 29	ax-mp 7	. . . 4 |- |^|{y e. On | (R1` U_x e. A suc (rank` x)) C_ (R1` y)} C_ |^|{y e. On | U_x e. A suc (rank` x) C_ y}
31		rankval2 5781	. . . . 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)) C_ (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)) C_ (R1` y)}
33		intmin 3237	. . . . . 6 |- (U_x e. A suc (rank` x) e. On -> |^|{y e. On | U_x e. A suc (rank` x) C_ 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) C_ y} = U_x e. A suc (rank` x)
35	34	eqcomi 1888	. . . 4 |- U_x e. A suc (rank` x) = |^|{y e. On | U_x e. A suc (rank` x) C_ y}
36	30, 32, 35	3sstr4i 2656	. . 3 |- (rank` (R1` U_x e. A suc (rank` x))) C_ U_x e. A suc (rank` x)
37	25, 36	sstri 2626	. 2 |- (rank` A) C_ U_x e. A suc (rank` x)
38		iunss 3291	. . 3 |- (U_x e. A suc (rank` x) C_ (rank` A) <-> A.x e. A suc (rank` x) C_ (rank` A))
39	11	rankel 5791	. . . 4 |- (x e. A -> (rank` x) e. (rank` A))
40		rankon 5782	. . . . 5 |- (rank` A) e. On
41	9, 40	onsucssi 3922	. . . 4 |- ((rank` x) e. (rank` A) <-> suc (rank` x) C_ (rank` A))
42	39, 41	sylib 215	. . 3 |- (x e. A -> suc (rank` x) C_ (rank` A))
43	38, 42	mprgbir 2163	. 2 |- U_x e. A suc (rank` x) C_ (rank` A)
44	37, 43	eqssi 2632	1 |- (rank` A) = U_x e. A suc (rank` x)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   C_ wss 2593  |^|cint 3214  U_ciun 3255  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankbnd 5814  rankc1 5816

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

Copyright terms: Public domain