Theorem rankval4 8288

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 2605	. . . . . 6 |- F/_ x A
2		nfcv 2605	. . . . . . 7 |- F/_ x R1
3		nfiu1 4345	. . . . . . 7 |- F/_ x U_ x e. A suc ( rank ` x )
4	2, 3	nffv 5863	. . . . . 6 |- F/_ x ( R1 ` U_ x e. A suc ( rank ` x ) )
5	1, 4	dfss2f 3480	. . . . 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 3098	. . . . . . 7 |- x e. _V
7	6	rankid 8254	. . . . . 6 |- x e. ( R1 ` suc ( rank ` x ) )
8		ssiun2 4358	. . . . . . . 8 |- ( x e. A -> suc ( rank ` x ) C_ U_ x e. A suc ( rank ` x ) )
9		rankon 8216	. . . . . . . . . 10 |- ( rank ` x ) e. On
10	9	onsuci 6658	. . . . . . . . 9 |- suc ( rank ` x ) e. On
11		rankr1b.1	. . . . . . . . . 10 |- A e. _V
12	10	rgenw 2804	. . . . . . . . . 10 |- A. x e. A suc ( rank ` x ) e. On
13		iunon 7011	. . . . . . . . . 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 672	. . . . . . . . 9 |- U_ x e. A suc ( rank ` x ) e. On
15		r1ord3 8203	. . . . . . . . 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 672	. . . . . . . 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 16	. . . . . . 7 |- ( x e. A -> ( R1 ` suc ( rank ` x ) ) C_ ( R1 ` U_ x e. A suc ( rank ` x ) ) )
18	17	sseld 3488	. . . . . 6 |- ( x e. A -> ( x e. ( R1 ` suc ( rank ` x ) ) -> x e. ( R1 ` U_ x e. A suc ( rank ` x ) ) ) )
19	7, 18	mpi 17	. . . . 5 |- ( x e. A -> x e. ( R1 ` U_ x e. A suc ( rank ` x ) ) )
20	5, 19	mpgbir 1609	. . . 4 |- A C_ ( R1 ` U_ x e. A suc ( rank ` x ) )
21		fvex 5866	. . . . 5 |- ( R1 ` U_ x e. A suc ( rank ` x ) ) e. _V
22	21	rankss 8270	. . . 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 8203	. . . . . . 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 670	. . . . . 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 3567	. . . . 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 4292	. . . . 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 8239	. . . . 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 4291	. . . . . 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 2456	. . . 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 3528	. . 3 |- ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) ) C_ U_ x e. A suc ( rank ` x )
35	23, 34	sstri 3498	. 2 |- ( rank ` A ) C_ U_ x e. A suc ( rank ` x )
36		iunss 4356	. . 3 |- ( U_ x e. A suc ( rank ` x ) C_ ( rank ` A ) <-> A. x e. A suc ( rank ` x ) C_ ( rank ` A ) )
37	11	rankel 8260	. . . 4 |- ( x e. A -> ( rank ` x ) e. ( rank ` A ) )
38		rankon 8216	. . . . 5 |- ( rank ` A ) e. On
39	9, 38	onsucssi 6661	. . . 4 |- ( ( rank ` x ) e. ( rank ` A ) <-> suc ( rank ` x ) C_ ( rank ` A ) )
40	37, 39	sylib 196	. . 3 |- ( x e. A -> suc ( rank ` x ) C_ ( rank ` A ) )
41	36, 40	mprgbir 2807	. 2 |- U_ x e. A suc ( rank ` x ) C_ ( rank ` A )
42	35, 41	eqssi 3505	1 |- ( rank ` A ) = U_ x e. A suc ( rank ` x )

Syntax hints:   -> wi 4   = wceq 1383   e. wcel 1804   A.wral 2793   {crab 2797   _Vcvv 3095   C_ wss 3461   |^|cint 4271   U_ciun 4315   Oncon0 4868   suc csuc 4870   ` cfv 5578   R1cr1 8183   rankcrnk 8184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078  df-r1 8185  df-rank 8186

This theorem is referenced by:  rankbnd  8289  rankc1  8291