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Theorem rankval4 7829
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.
StepHypRef Expression
1 nfcv 2579 . . . . . 6  |-  F/_ x A
2 nfcv 2579 . . . . . . 7  |-  F/_ x R1
3 nfiu1 4151 . . . . . . 7  |-  F/_ x U_ x  e.  A  suc  ( rank `  x
)
42, 3nffv 5770 . . . . . 6  |-  F/_ x
( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
51, 4dfss2f 3328 . . . . 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 2968 . . . . . . 7  |-  x  e. 
_V
76rankid 7795 . . . . . 6  |-  x  e.  ( R1 `  suc  ( rank `  x )
)
8 ssiun2 4164 . . . . . . . 8  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_ 
U_ x  e.  A  suc  ( rank `  x
) )
9 rankon 7757 . . . . . . . . . 10  |-  ( rank `  x )  e.  On
109onsuci 4853 . . . . . . . . 9  |-  suc  ( rank `  x )  e.  On
11 rankr1b.1 . . . . . . . . . 10  |-  A  e. 
_V
1210rgenw 2780 . . . . . . . . . 10  |-  A. x  e.  A  suc  ( rank `  x )  e.  On
13 iunon 6636 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A. x  e.  A  suc  ( rank `  x )  e.  On )  ->  U_ x  e.  A  suc  ( rank `  x )  e.  On )
1411, 12, 13mp2an 655 . . . . . . . . 9  |-  U_ x  e.  A  suc  ( rank `  x )  e.  On
15 r1ord3 7744 . . . . . . . . 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 ) ) ) )
1610, 14, 15mp2an 655 . . . . . . . 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 ) ) )
178, 16syl 16 . . . . . . 7  |-  ( x  e.  A  ->  ( R1 `  suc  ( rank `  x ) )  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) )
1817sseld 3336 . . . . . 6  |-  ( x  e.  A  ->  (
x  e.  ( R1
`  suc  ( rank `  x ) )  ->  x  e.  ( R1 ` 
U_ x  e.  A  suc  ( rank `  x
) ) ) )
197, 18mpi 17 . . . . 5  |-  ( x  e.  A  ->  x  e.  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) )
205, 19mpgbir 1560 . . . 4  |-  A  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
21 fvex 5773 . . . . 5  |-  ( R1
`  U_ x  e.  A  suc  ( rank `  x
) )  e.  _V
2221rankss 7811 . . . 4  |-  ( A 
C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x )
)  ->  ( rank `  A )  C_  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x )
) ) )
2320, 22ax-mp 5 . . 3  |-  ( rank `  A )  C_  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x )
) )
24 r1ord3 7744 . . . . . . 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 ) ) )
2514, 24mpan 653 . . . . . 6  |-  ( y  e.  On  ->  ( U_ x  e.  A  suc  ( rank `  x
)  C_  y  ->  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) )  C_  ( R1 `  y ) ) )
2625ss2rabi 3414 . . . . 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 4100 . . . . 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 }
)
2826, 27ax-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 7780 . . . . 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 ) } )
3021, 29ax-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 4099 . . . . . 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 )
)
3214, 31ax-mp 5 . . . . 5  |-  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x )  C_  y }  =  U_ x  e.  A  suc  ( rank `  x )
3332eqcomi 2447 . . . 4  |-  U_ x  e.  A  suc  ( rank `  x )  =  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x
)  C_  y }
3428, 30, 333sstr4i 3376 . . 3  |-  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) ) 
C_  U_ x  e.  A  suc  ( rank `  x
)
3523, 34sstri 3346 . 2  |-  ( rank `  A )  C_  U_ x  e.  A  suc  ( rank `  x )
36 iunss 4162 . . 3  |-  ( U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A
)  <->  A. x  e.  A  suc  ( rank `  x
)  C_  ( rank `  A ) )
3711rankel 7801 . . . 4  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
38 rankon 7757 . . . . 5  |-  ( rank `  A )  e.  On
399, 38onsucssi 4856 . . . 4  |-  ( (
rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  C_  ( rank `  A ) )
4037, 39sylib 190 . . 3  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_  ( rank `  A
) )
4136, 40mprgbir 2783 . 2  |-  U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A )
4235, 41eqssi 3353 1  |-  ( rank `  A )  =  U_ x  e.  A  suc  ( rank `  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1654    e. wcel 1728   A.wral 2712   {crab 2716   _Vcvv 2965    C_ wss 3309   |^|cint 4079   U_ciun 4122   Oncon0 4616   suc csuc 4618   ` cfv 5489   R1cr1 7724   rankcrnk 7725
This theorem is referenced by:  rankbnd  7830  rankc1  7832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-reg 7596  ax-inf2 7632
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-recs 6669  df-rdg 6704  df-r1 7726  df-rank 7727
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