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Theorem rankval4 8276
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 2624 . . . . . 6  |-  F/_ x A
2 nfcv 2624 . . . . . . 7  |-  F/_ x R1
3 nfiu1 4350 . . . . . . 7  |-  F/_ x U_ x  e.  A  suc  ( rank `  x
)
42, 3nffv 5866 . . . . . 6  |-  F/_ x
( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
51, 4dfss2f 3490 . . . . 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 3111 . . . . . . 7  |-  x  e. 
_V
76rankid 8242 . . . . . 6  |-  x  e.  ( R1 `  suc  ( rank `  x )
)
8 ssiun2 4363 . . . . . . . 8  |-  ( x  e.  A  ->  suc  ( rank `  x )  C_ 
U_ x  e.  A  suc  ( rank `  x
) )
9 rankon 8204 . . . . . . . . . 10  |-  ( rank `  x )  e.  On
109onsuci 6646 . . . . . . . . 9  |-  suc  ( rank `  x )  e.  On
11 rankr1b.1 . . . . . . . . . 10  |-  A  e. 
_V
1210rgenw 2820 . . . . . . . . . 10  |-  A. x  e.  A  suc  ( rank `  x )  e.  On
13 iunon 7001 . . . . . . . . . 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 8191 . . . . . . . . 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 3498 . . . . . 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 1600 . . . 4  |-  A  C_  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) )
21 fvex 5869 . . . . 5  |-  ( R1
`  U_ x  e.  A  suc  ( rank `  x
) )  e.  _V
2221rankss 8258 . . . 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 8191 . . . . . . 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 3577 . . . . 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 4298 . . . . 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 8227 . . . . 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 4297 . . . . . 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 2475 . . . 4  |-  U_ x  e.  A  suc  ( rank `  x )  =  |^| { y  e.  On  |  U_ x  e.  A  suc  ( rank `  x
)  C_  y }
3428, 30, 333sstr4i 3538 . . 3  |-  ( rank `  ( R1 `  U_ x  e.  A  suc  ( rank `  x ) ) ) 
C_  U_ x  e.  A  suc  ( rank `  x
)
3523, 34sstri 3508 . 2  |-  ( rank `  A )  C_  U_ x  e.  A  suc  ( rank `  x )
36 iunss 4361 . . 3  |-  ( U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A
)  <->  A. x  e.  A  suc  ( rank `  x
)  C_  ( rank `  A ) )
3711rankel 8248 . . . 4  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
38 rankon 8204 . . . . 5  |-  ( rank `  A )  e.  On
399, 38onsucssi 6649 . . . 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 2823 . 2  |-  U_ x  e.  A  suc  ( rank `  x )  C_  ( rank `  A )
4235, 41eqssi 3515 1  |-  ( rank `  A )  =  U_ x  e.  A  suc  ( rank `  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813   _Vcvv 3108    C_ wss 3471   |^|cint 4277   U_ciun 4320   Oncon0 4873   suc csuc 4875   ` cfv 5581   R1cr1 8171   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-reg 8009  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by:  rankbnd  8277  rankc1  8279
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