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Theorem rankval4 7535
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem rankval4
StepHypRef Expression
1 nfcv 2421 . . . . . 6  |-  F/_ x A
2 nfcv 2421 . . . . . . 7  |-  F/_ x R1
3 nfiu1 3935 . . . . . . 7  |-  F/_ x U_ x  e.  A  suc  ( rank `  x
)
42, 3nffv 5493 . . . . . 6  |-  F/_ x
( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
51, 4dfss2f 3173 . . . . 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 2793 . . . . . . 7  |-  x  e. 
_V
76rankid 7501 . . . . . 6  |-  x  e.  ( R1 `  suc  ( rank `  x )
)
8 ssiun2 3947 . . . . . . . 8  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_ 
U_ x  e.  A  suc  ( rank `  x
) )
9 rankon 7463 . . . . . . . . . 10  |-  ( rank `  x )  e.  On
109onsuci 4629 . . . . . . . . 9  |-  suc  ( rank `  x )  e.  On
11 rankr1b.1 . . . . . . . . . 10  |-  A  e. 
_V
1210rgenw 2612 . . . . . . . . . 10  |-  A. x  e.  A  suc  ( rank `  x )  e.  On
13 iunon 6351 . . . . . . . . . 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 7450 . . . . . . . . 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 17 . . . . . . 7  |-  ( x  e.  A  ->  ( R1 `  suc  ( rank `  x ) )  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) )
1817sseld 3181 . . . . . 6  |-  ( x  e.  A  ->  (
x  e.  ( R1
`  suc  ( rank `  x ) )  ->  x  e.  ( R1 ` 
U_ x  e.  A  suc  ( rank `  x
) ) ) )
197, 18mpi 18 . . . . 5  |-  ( x  e.  A  ->  x  e.  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) )
205, 19mpgbir 1538 . . . 4  |-  A  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
21 fvex 5500 . . . . 5  |-  ( R1
`  U_ x  e.  A  suc  ( rank `  x
) )  e.  _V
2221rankss 7517 . . . 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 10 . . 3  |-  ( rank `  A )  C_  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x )
) )
24 r1ord3 7450 . . . . . . 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 3257 . . . . 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 3885 . . . . 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 10 . . . 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 7486 . . . . 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 10 . . . 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 3884 . . . . . 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 10 . . . . 5  |-  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x )  C_  y }  =  U_ x  e.  A  suc  ( rank `  x )
3332eqcomi 2289 . . . 4  |-  U_ x  e.  A  suc  ( rank `  x )  =  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x
)  C_  y }
3428, 30, 333sstr4i 3219 . . 3  |-  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) ) 
C_  U_ x  e.  A  suc  ( rank `  x
)
3523, 34sstri 3190 . 2  |-  ( rank `  A )  C_  U_ x  e.  A  suc  ( rank `  x )
36 iunss 3945 . . 3  |-  ( U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A
)  <->  A. x  e.  A  suc  ( rank `  x
)  C_  ( rank `  A ) )
3711rankel 7507 . . . 4  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
38 rankon 7463 . . . . 5  |-  ( rank `  A )  e.  On
399, 38onsucssi 4632 . . . 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 2615 . 2  |-  U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A )
4235, 41eqssi 3197 1  |-  ( rank `  A )  =  U_ x  e.  A  suc  ( rank `  x )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   A.wral 2545   {crab 2549   _Vcvv 2790    C_ wss 3154   |^|cint 3864   U_ciun 3907   Oncon0 4392   suc csuc 4394   ` cfv 5222   R1cr1 7430   rankcrnk 7431
This theorem is referenced by:  rankbnd  7536  rankc1  7538
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7302  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384  df-rdg 6419  df-r1 7432  df-rank 7433
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