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Theorem bndrank 7603
Description: Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
bndrank  |-  ( E. x  e.  On  A. y  e.  A  ( rank `  y )  C_  x  ->  A  e.  _V )
Distinct variable group:    x, y, A

Proof of Theorem bndrank
StepHypRef Expression
1 rankon 7557 . . . . . . . 8  |-  ( rank `  y )  e.  On
21onordi 4579 . . . . . . 7  |-  Ord  ( rank `  y )
3 eloni 4484 . . . . . . 7  |-  ( x  e.  On  ->  Ord  x )
4 ordsucsssuc 4696 . . . . . . 7  |-  ( ( Ord  ( rank `  y
)  /\  Ord  x )  ->  ( ( rank `  y )  C_  x  <->  suc  ( rank `  y
)  C_  suc  x ) )
52, 3, 4sylancr 644 . . . . . 6  |-  ( x  e.  On  ->  (
( rank `  y )  C_  x  <->  suc  ( rank `  y
)  C_  suc  x ) )
61onsuci 4711 . . . . . . 7  |-  suc  ( rank `  y )  e.  On
7 suceloni 4686 . . . . . . 7  |-  ( x  e.  On  ->  suc  x  e.  On )
8 r1ord3 7544 . . . . . . 7  |-  ( ( suc  ( rank `  y
)  e.  On  /\  suc  x  e.  On )  ->  ( suc  ( rank `  y )  C_  suc  x  ->  ( R1 ` 
suc  ( rank `  y
) )  C_  ( R1 `  suc  x ) ) )
96, 7, 8sylancr 644 . . . . . 6  |-  ( x  e.  On  ->  ( suc  ( rank `  y
)  C_  suc  x  -> 
( R1 `  suc  ( rank `  y )
)  C_  ( R1 ` 
suc  x ) ) )
105, 9sylbid 206 . . . . 5  |-  ( x  e.  On  ->  (
( rank `  y )  C_  x  ->  ( R1 ` 
suc  ( rank `  y
) )  C_  ( R1 `  suc  x ) ) )
11 vex 2867 . . . . . 6  |-  y  e. 
_V
1211rankid 7595 . . . . 5  |-  y  e.  ( R1 `  suc  ( rank `  y )
)
13 ssel 3250 . . . . 5  |-  ( ( R1 `  suc  ( rank `  y ) ) 
C_  ( R1 `  suc  x )  ->  (
y  e.  ( R1
`  suc  ( rank `  y ) )  -> 
y  e.  ( R1
`  suc  x )
) )
1410, 12, 13syl6mpi 58 . . . 4  |-  ( x  e.  On  ->  (
( rank `  y )  C_  x  ->  y  e.  ( R1 `  suc  x
) ) )
1514ralimdv 2698 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  A  ( rank `  y )  C_  x  ->  A. y  e.  A  y  e.  ( R1 `  suc  x
) ) )
16 dfss3 3246 . . . 4  |-  ( A 
C_  ( R1 `  suc  x )  <->  A. y  e.  A  y  e.  ( R1 `  suc  x
) )
17 fvex 5622 . . . . 5  |-  ( R1
`  suc  x )  e.  _V
1817ssex 4239 . . . 4  |-  ( A 
C_  ( R1 `  suc  x )  ->  A  e.  _V )
1916, 18sylbir 204 . . 3  |-  ( A. y  e.  A  y  e.  ( R1 `  suc  x )  ->  A  e.  _V )
2015, 19syl6 29 . 2  |-  ( x  e.  On  ->  ( A. y  e.  A  ( rank `  y )  C_  x  ->  A  e.  _V ) )
2120rexlimiv 2737 1  |-  ( E. x  e.  On  A. y  e.  A  ( rank `  y )  C_  x  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1710   A.wral 2619   E.wrex 2620   _Vcvv 2864    C_ wss 3228   Ord word 4473   Oncon0 4474   suc csuc 4476   ` cfv 5337   R1cr1 7524   rankcrnk 7525
This theorem is referenced by:  unbndrank  7604  scottex  7645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-reg 7396  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-r1 7526  df-rank 7527
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