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Theorem rankval2 7506
Description: Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.)
Assertion
Ref Expression
rankval2  |-  ( A  e.  B  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  C_  ( R1
`  x ) } )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem rankval2
StepHypRef Expression
1 rankvalg 7505 . 2  |-  ( A  e.  B  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
2 r1suc 7458 . . . . . 6  |-  ( x  e.  On  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
32eleq2d 2363 . . . . 5  |-  ( x  e.  On  ->  ( A  e.  ( R1 ` 
suc  x )  <->  A  e.  ~P ( R1 `  x
) ) )
4 fvex 5555 . . . . . 6  |-  ( R1
`  x )  e. 
_V
54elpw2 4191 . . . . 5  |-  ( A  e.  ~P ( R1
`  x )  <->  A  C_  ( R1 `  x ) )
63, 5syl6bb 252 . . . 4  |-  ( x  e.  On  ->  ( A  e.  ( R1 ` 
suc  x )  <->  A  C_  ( R1 `  x ) ) )
76rabbiia 2791 . . 3  |-  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  =  { x  e.  On  |  A  C_  ( R1
`  x ) }
87inteqi 3882 . 2  |-  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  =  |^| { x  e.  On  |  A  C_  ( R1
`  x ) }
91, 8syl6eq 2344 1  |-  ( A  e.  B  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  C_  ( R1
`  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   ~Pcpw 3638   |^|cint 3878   Oncon0 4408   suc csuc 4410   ` cfv 5271   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  rankval4  7555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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