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

Theorem rankbnd2 8287
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 8281 . . . . 5  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
2 rankr1b.1 . . . . . 6  |-  A  e. 
_V
32rankuni2 8273 . . . . 5  |-  ( rank `  U. A )  = 
U_ x  e.  A  ( rank `  x )
41, 3eqtr3i 2498 . . . 4  |-  U. ( rank `  A )  = 
U_ x  e.  A  ( rank `  x )
54sseq1i 3528 . . 3  |-  ( U. ( rank `  A )  C_  B  <->  U_ x  e.  A  ( rank `  x )  C_  B )
6 iunss 4366 . . 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 8213 . . . 4  |-  ( rank `  A )  e.  On
98onssi 6656 . . 3  |-  ( rank `  A )  C_  On
10 eloni 4888 . . 3  |-  ( B  e.  On  ->  Ord  B )
11 ordunisssuc 4980 . . 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 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   U.cuni 4245   U_ciun 4325   Ord word 4877   Oncon0 4878   suc csuc 4880   ` cfv 5588   rankcrnk 8181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-recs 7042  df-rdg 7076  df-r1 8182  df-rank 8183
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator