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Theorem rankval4 7782
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 2571 . . . . . 6  |-  F/_ x A
2 nfcv 2571 . . . . . . 7  |-  F/_ x R1
3 nfiu1 4113 . . . . . . 7  |-  F/_ x U_ x  e.  A  suc  ( rank `  x
)
42, 3nffv 5726 . . . . . 6  |-  F/_ x
( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
51, 4dfss2f 3331 . . . . 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 2951 . . . . . . 7  |-  x  e. 
_V
76rankid 7748 . . . . . 6  |-  x  e.  ( R1 `  suc  ( rank `  x )
)
8 ssiun2 4126 . . . . . . . 8  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_ 
U_ x  e.  A  suc  ( rank `  x
) )
9 rankon 7710 . . . . . . . . . 10  |-  ( rank `  x )  e.  On
109onsuci 4809 . . . . . . . . 9  |-  suc  ( rank `  x )  e.  On
11 rankr1b.1 . . . . . . . . . 10  |-  A  e. 
_V
1210rgenw 2765 . . . . . . . . . 10  |-  A. x  e.  A  suc  ( rank `  x )  e.  On
13 iunon 6591 . . . . . . . . . 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 654 . . . . . . . . 9  |-  U_ x  e.  A  suc  ( rank `  x )  e.  On
15 r1ord3 7697 . . . . . . . . 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 654 . . . . . . . 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 3339 . . . . . 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 1559 . . . 4  |-  A  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
21 fvex 5733 . . . . 5  |-  ( R1
`  U_ x  e.  A  suc  ( rank `  x
) )  e.  _V
2221rankss 7764 . . . 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 8 . . 3  |-  ( rank `  A )  C_  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x )
) )
24 r1ord3 7697 . . . . . . 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 652 . . . . . 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 3417 . . . . 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 4063 . . . . 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 8 . . . 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 7733 . . . . 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 8 . . . 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 4062 . . . . . 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 8 . . . . 5  |-  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x )  C_  y }  =  U_ x  e.  A  suc  ( rank `  x )
3332eqcomi 2439 . . . 4  |-  U_ x  e.  A  suc  ( rank `  x )  =  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x
)  C_  y }
3428, 30, 333sstr4i 3379 . . 3  |-  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) ) 
C_  U_ x  e.  A  suc  ( rank `  x
)
3523, 34sstri 3349 . 2  |-  ( rank `  A )  C_  U_ x  e.  A  suc  ( rank `  x )
36 iunss 4124 . . 3  |-  ( U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A
)  <->  A. x  e.  A  suc  ( rank `  x
)  C_  ( rank `  A ) )
3711rankel 7754 . . . 4  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
38 rankon 7710 . . . . 5  |-  ( rank `  A )  e.  On
399, 38onsucssi 4812 . . . 4  |-  ( (
rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  C_  ( rank `  A ) )
4037, 39sylib 189 . . 3  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_  ( rank `  A
) )
4136, 40mprgbir 2768 . 2  |-  U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A )
4235, 41eqssi 3356 1  |-  ( rank `  A )  =  U_ x  e.  A  suc  ( rank `  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    C_ wss 3312   |^|cint 4042   U_ciun 4085   Oncon0 4573   suc csuc 4575   ` cfv 5445   R1cr1 7677   rankcrnk 7678
This theorem is referenced by:  rankbnd  7783  rankc1  7785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-reg 7549  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-recs 6624  df-rdg 6659  df-r1 7679  df-rank 7680
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