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

Theorem rankbnd2 8304
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 8298 . . . . 5  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
2 rankr1b.1 . . . . . 6  |-  A  e. 
_V
32rankuni2 8290 . . . . 5  |-  ( rank `  U. A )  = 
U_ x  e.  A  ( rank `  x )
41, 3eqtr3i 2488 . . . 4  |-  U. ( rank `  A )  = 
U_ x  e.  A  ( rank `  x )
54sseq1i 3523 . . 3  |-  ( U. ( rank `  A )  C_  B  <->  U_ x  e.  A  ( rank `  x )  C_  B )
6 iunss 4373 . . 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 8230 . . . 4  |-  ( rank `  A )  e.  On
98onssi 6671 . . 3  |-  ( rank `  A )  C_  On
10 eloni 4897 . . 3  |-  ( B  e.  On  ->  Ord  B )
11 ordunisssuc 4989 . . 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 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   U.cuni 4251   U_ciun 4332   Ord word 4886   Oncon0 4887   suc csuc 4889   ` cfv 5594   rankcrnk 8198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator