Theorem rankval4 8730

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. (Contributed by NM, 12-Oct-2003.)

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

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . . . 6 |- F/_ x A
2		nfcv 2764	. . . . . . 7 |- F/_ x R1
3		nfiu1 4550	. . . . . . 7 |- F/_ x U_ x e. A suc ( rank ` x )
4	2, 3	nffv 6198	. . . . . 6 |- F/_ x ( R1 ` U_ x e. A suc ( rank ` x ) )
5	1, 4	dfss2f 3594	. . . . 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		vex 3203	. . . . . . 7 |- x e. _V
7	6	rankid 8696	. . . . . 6 |- x e. ( R1 ` suc ( rank ` x ) )
8		ssiun2 4563	. . . . . . . 8 |- ( x e. A -> suc ( rank ` x ) C_ U_ x e. A suc ( rank ` x ) )
9		rankon 8658	. . . . . . . . . 10 |- ( rank ` x ) e. On
10	9	onsuci 7038	. . . . . . . . 9 |- suc ( rank ` x ) e. On
11		rankr1b.1	. . . . . . . . . 10 |- A e. _V
12	10	rgenw 2924	. . . . . . . . . 10 |- A. x e. A suc ( rank ` x ) e. On
13		iunon 7436	. . . . . . . . . 10 |- ( ( A e. _V /\ A. x e. A suc ( rank ` x ) e. On ) -> U_ x e. A suc ( rank ` x ) e. On )
14	11, 12, 13	mp2an 708	. . . . . . . . 9 |- U_ x e. A suc ( rank ` x ) e. On
15		r1ord3 8645	. . . . . . . . 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 ) ) ) )
16	10, 14, 15	mp2an 708	. . . . . . . 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 ) ) )
17	8, 16	syl 17	. . . . . . 7 |- ( x e. A -> ( R1 ` suc ( rank ` x ) ) C_ ( R1 ` U_ x e. A suc ( rank ` x ) ) )
18	17	sseld 3602	. . . . . 6 |- ( x e. A -> ( x e. ( R1 ` suc ( rank ` x ) ) -> x e. ( R1 ` U_ x e. A suc ( rank ` x ) ) ) )
19	7, 18	mpi 20	. . . . 5 |- ( x e. A -> x e. ( R1 ` U_ x e. A suc ( rank ` x ) ) )
20	5, 19	mpgbir 1726	. . . 4 |- A C_ ( R1 ` U_ x e. A suc ( rank ` x ) )
21		fvex 6201	. . . . 5 |- ( R1 ` U_ x e. A suc ( rank ` x ) ) e. _V
22	21	rankss 8712	. . . 4 |- ( A C_ ( R1 ` U_ x e. A suc ( rank ` x ) ) -> ( rank ` A ) C_ ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) ) )
23	20, 22	ax-mp 5	. . 3 |- ( rank ` A ) C_ ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) )
24		r1ord3 8645	. . . . . . 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 ) ) )
25	14, 24	mpan 706	. . . . . 6 |- ( y e. On -> ( U_ x e. A suc ( rank ` x ) C_ y -> ( R1 ` U_ x e. A suc ( rank ` x ) ) C_ ( R1 ` y ) ) )
26	25	ss2rabi 3684	. . . . 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 ) }
27		intss 4498	. . . . 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 } )
28	26, 27	ax-mp 5	. . . 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 }
29		rankval2 8681	. . . . 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 ) } )
30	21, 29	ax-mp 5	. . . 4 |- ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) ) = |^| { y e. On | ( R1 ` U_ x e. A suc ( rank ` x ) ) C_ ( R1 ` y ) }
31		intmin 4497	. . . . . 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 ) )
32	14, 31	ax-mp 5	. . . . 5 |- |^| { y e. On | U_ x e. A suc ( rank ` x ) C_ y } = U_ x e. A suc ( rank ` x )
33	32	eqcomi 2631	. . . 4 |- U_ x e. A suc ( rank ` x ) = |^| { y e. On | U_ x e. A suc ( rank ` x ) C_ y }
34	28, 30, 33	3sstr4i 3644	. . 3 |- ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) ) C_ U_ x e. A suc ( rank ` x )
35	23, 34	sstri 3612	. 2 |- ( rank ` A ) C_ U_ x e. A suc ( rank ` x )
36		iunss 4561	. . 3 |- ( U_ x e. A suc ( rank ` x ) C_ ( rank ` A ) <-> A. x e. A suc ( rank ` x ) C_ ( rank ` A ) )
37	11	rankel 8702	. . . 4 |- ( x e. A -> ( rank ` x ) e. ( rank ` A ) )
38		rankon 8658	. . . . 5 |- ( rank ` A ) e. On
39	9, 38	onsucssi 7041	. . . 4 |- ( ( rank ` x ) e. ( rank ` A ) <-> suc ( rank ` x ) C_ ( rank ` A ) )
40	37, 39	sylib 208	. . 3 |- ( x e. A -> suc ( rank ` x ) C_ ( rank ` A ) )
41	36, 40	mprgbir 2927	. 2 |- U_ x e. A suc ( rank ` x ) C_ ( rank ` A )
42	35, 41	eqssi 3619	1 |- ( rank ` A ) = U_ x e. A suc ( rank ` x )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   |^|cint 4475   U_ciun 4520   Oncon0 5723   suc csuc 5725   ` cfv 5888   R1cr1 8625   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  rankbnd  8731  rankc1  8733