Theorem rankval4 8276

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 2624	. . . . . 6 |- F/_ x A
2		nfcv 2624	. . . . . . 7 |- F/_ x R1
3		nfiu1 4350	. . . . . . 7 |- F/_ x U_ x e. A suc ( rank ` x )
4	2, 3	nffv 5866	. . . . . 6 |- F/_ x ( R1 ` U_ x e. A suc ( rank ` x ) )
5	1, 4	dfss2f 3490	. . . . 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 3111	. . . . . . 7 |- x e. _V
7	6	rankid 8242	. . . . . 6 |- x e. ( R1 ` suc ( rank ` x ) )
8		ssiun2 4363	. . . . . . . 8 |- ( x e. A -> suc ( rank ` x ) C_ U_ x e. A suc ( rank ` x ) )
9		rankon 8204	. . . . . . . . . 10 |- ( rank ` x ) e. On
10	9	onsuci 6646	. . . . . . . . 9 |- suc ( rank ` x ) e. On
11		rankr1b.1	. . . . . . . . . 10 |- A e. _V
12	10	rgenw 2820	. . . . . . . . . 10 |- A. x e. A suc ( rank ` x ) e. On
13		iunon 7001	. . . . . . . . . 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 8191	. . . . . . . . 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 3498	. . . . . 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 1600	. . . 4 |- A C_ ( R1 ` U_ x e. A suc ( rank ` x ) )
21		fvex 5869	. . . . 5 |- ( R1 ` U_ x e. A suc ( rank ` x ) ) e. _V
22	21	rankss 8258	. . . 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 8191	. . . . . . 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 3577	. . . . 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 4298	. . . . 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 8227	. . . . 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 4297	. . . . . 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 2475	. . . 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 3538	. . 3 |- ( rank ` ( R1 ` U_ x e. A suc ( rank ` x ) ) ) C_ U_ x e. A suc ( rank ` x )
35	23, 34	sstri 3508	. 2 |- ( rank ` A ) C_ U_ x e. A suc ( rank ` x )
36		iunss 4361	. . 3 |- ( U_ x e. A suc ( rank ` x ) C_ ( rank ` A ) <-> A. x e. A suc ( rank ` x ) C_ ( rank ` A ) )
37	11	rankel 8248	. . . 4 |- ( x e. A -> ( rank ` x ) e. ( rank ` A ) )
38		rankon 8204	. . . . 5 |- ( rank ` A ) e. On
39	9, 38	onsucssi 6649	. . . 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 2823	. 2 |- U_ x e. A suc ( rank ` x ) C_ ( rank ` A )
42	35, 41	eqssi 3515	1 |- ( rank ` A ) = U_ x e. A suc ( rank ` x )

Syntax hints:   -> wi 4   = wceq 1374   e. wcel 1762   A.wral 2809   {crab 2813   _Vcvv 3108   C_ wss 3471   |^|cint 4277   U_ciun 4320   Oncon0 4873   suc csuc 4875   ` cfv 5581   R1cr1 8171   rankcrnk 8172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-reg 8009  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174

This theorem is referenced by:  rankbnd  8277  rankc1  8279