Theorem rankval4 8363

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 2602	. . . . . 6 |- F/_ x A
2		nfcv 2602	. . . . . . 7 |- F/_ x R1
3		nfiu1 4321	. . . . . . 7 |- F/_ x U_ x e. A suc ( rank ` x )
4	2, 3	nffv 5894	. . . . . 6 |- F/_ x ( R1 ` U_ x e. A suc ( rank ` x ) )
5	1, 4	dfss2f 3434	. . . . 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 3059	. . . . . . 7 |- x e. _V
7	6	rankid 8329	. . . . . 6 |- x e. ( R1 ` suc ( rank ` x ) )
8		ssiun2 4334	. . . . . . . 8 |- ( x e. A -> suc ( rank ` x ) C_ U_ x e. A suc ( rank ` x ) )
9		rankon 8291	. . . . . . . . . 10 |- ( rank ` x ) e. On
10	9	onsuci 6691	. . . . . . . . 9 |- suc ( rank ` x ) e. On
11		rankr1b.1	. . . . . . . . . 10 |- A e. _V
12	10	rgenw 2760	. . . . . . . . . 10 |- A. x e. A suc ( rank ` x ) e. On
13		iunon 7082	. . . . . . . . . 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 683	. . . . . . . . 9 |- U_ x e. A suc ( rank ` x ) e. On
15		r1ord3 8278	. . . . . . . . 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 683	. . . . . . . 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 3442	. . . . . 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 1683	. . . 4 |- A C_ ( R1 ` U_ x e. A suc ( rank ` x ) )
21		fvex 5897	. . . . 5 |- ( R1 ` U_ x e. A suc ( rank ` x ) ) e. _V
22	21	rankss 8345	. . . 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 8278	. . . . . . 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 681	. . . . . 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 3522	. . . . 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 4268	. . . . 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 8314	. . . . 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 4267	. . . . . 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 2470	. . . 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 3482	. . 3 |- ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) ) C_ U_ x e. A suc ( rank ` x )
35	23, 34	sstri 3452	. 2 |- ( rank ` A ) C_ U_ x e. A suc ( rank ` x )
36		iunss 4332	. . 3 |- ( U_ x e. A suc ( rank ` x ) C_ ( rank ` A ) <-> A. x e. A suc ( rank ` x ) C_ ( rank ` A ) )
37	11	rankel 8335	. . . 4 |- ( x e. A -> ( rank ` x ) e. ( rank ` A ) )
38		rankon 8291	. . . . 5 |- ( rank ` A ) e. On
39	9, 38	onsucssi 6694	. . . 4 |- ( ( rank ` x ) e. ( rank ` A ) <-> suc ( rank ` x ) C_ ( rank ` A ) )
40	37, 39	sylib 201	. . 3 |- ( x e. A -> suc ( rank ` x ) C_ ( rank ` A ) )
41	36, 40	mprgbir 2763	. 2 |- U_ x e. A suc ( rank ` x ) C_ ( rank ` A )
42	35, 41	eqssi 3459	1 |- ( rank ` A ) = U_ x e. A suc ( rank ` x )

Syntax hints:   -> wi 4   = wceq 1454   e. wcel 1897   A.wral 2748   {crab 2752   _Vcvv 3056   C_ wss 3415   |^|cint 4247   U_ciun 4291   Oncon0 5441   suc csuc 5443   ` cfv 5600   R1cr1 8258   rankcrnk 8259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-reg 8132  ax-inf2 8171

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-r1 8260  df-rank 8261

This theorem is referenced by:  rankbnd  8364  rankc1  8366