MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankbnd2 Structured version   Unicode version

Theorem rankbnd2 8068
Description: The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.)
Hypothesis
Ref Expression
rankr1b.1  |-  A  e. 
_V
Assertion
Ref Expression
rankbnd2  |-  ( B  e.  On  ->  ( A. x  e.  A  ( rank `  x )  C_  B  <->  ( rank `  A
)  C_  suc  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem rankbnd2
StepHypRef Expression
1 rankuni 8062 . . . . 5  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
2 rankr1b.1 . . . . . 6  |-  A  e. 
_V
32rankuni2 8054 . . . . 5  |-  ( rank `  U. A )  = 
U_ x  e.  A  ( rank `  x )
41, 3eqtr3i 2460 . . . 4  |-  U. ( rank `  A )  = 
U_ x  e.  A  ( rank `  x )
54sseq1i 3375 . . 3  |-  ( U. ( rank `  A )  C_  B  <->  U_ x  e.  A  ( rank `  x )  C_  B )
6 iunss 4206 . . 3  |-  ( U_ x  e.  A  ( rank `  x )  C_  B 
<-> 
A. x  e.  A  ( rank `  x )  C_  B )
75, 6bitr2i 250 . 2  |-  ( A. x  e.  A  ( rank `  x )  C_  B 
<-> 
U. ( rank `  A
)  C_  B )
8 rankon 7994 . . . 4  |-  ( rank `  A )  e.  On
98onssi 6443 . . 3  |-  ( rank `  A )  C_  On
10 eloni 4724 . . 3  |-  ( B  e.  On  ->  Ord  B )
11 ordunisssuc 4816 . . 3  |-  ( ( ( rank `  A
)  C_  On  /\  Ord  B )  ->  ( U. ( rank `  A )  C_  B  <->  ( rank `  A
)  C_  suc  B ) )
129, 10, 11sylancr 663 . 2  |-  ( B  e.  On  ->  ( U. ( rank `  A
)  C_  B  <->  ( rank `  A )  C_  suc  B ) )
137, 12syl5bb 257 1  |-  ( B  e.  On  ->  ( A. x  e.  A  ( rank `  x )  C_  B  <->  ( rank `  A
)  C_  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1756   A.wral 2710   _Vcvv 2967    C_ wss 3323   U.cuni 4086   U_ciun 4166   Ord word 4713   Oncon0 4714   suc csuc 4716   ` cfv 5413   rankcrnk 7962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-reg 7799  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator