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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.
StepHypRef 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
)
42, 3nffv 5863 . . . . . 6  |-  F/_ x
( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
51, 4dfss2f 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
76rankid 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
109onsuci 6658 . . . . . . . . 9  |-  suc  ( rank `  x )  e.  On
11 rankr1b.1 . . . . . . . . . 10  |-  A  e. 
_V
1210rgenw 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 )
1411, 12, 13mp2an 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 ) ) ) )
1610, 14, 15mp2an 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 ) ) )
178, 16syl 16 . . . . . . 7  |-  ( x  e.  A  ->  ( R1 `  suc  ( rank `  x ) )  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) )
1817sseld 3488 . . . . . 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 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
2221rankss 8270 . . . 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 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 ) ) )
2514, 24mpan 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 ) ) )
2625ss2rabi 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 }
)
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 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 ) } )
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 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 )
)
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 2456 . . . 4  |-  U_ x  e.  A  suc  ( rank `  x )  =  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x
)  C_  y }
3428, 30, 333sstr4i 3528 . . 3  |-  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) ) 
C_  U_ x  e.  A  suc  ( rank `  x
)
3523, 34sstri 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 ) )
3711rankel 8260 . . . 4  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
38 rankon 8216 . . . . 5  |-  ( rank `  A )  e.  On
399, 38onsucssi 6661 . . . 4  |-  ( (
rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  C_  ( rank `  A ) )
4037, 39sylib 196 . . 3  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_  ( rank `  A
) )
4136, 40mprgbir 2807 . 2  |-  U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A )
4235, 41eqssi 3505 1  |-  ( rank `  A )  =  U_ x  e.  A  suc  ( rank `  x )
Colors of variables: wff setvar class
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
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