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Theorem rankbnd2 8186
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 8180 . . . . 5  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
2 rankr1b.1 . . . . . 6  |-  A  e. 
_V
32rankuni2 8172 . . . . 5  |-  ( rank `  U. A )  = 
U_ x  e.  A  ( rank `  x )
41, 3eqtr3i 2485 . . . 4  |-  U. ( rank `  A )  = 
U_ x  e.  A  ( rank `  x )
54sseq1i 3487 . . 3  |-  ( U. ( rank `  A )  C_  B  <->  U_ x  e.  A  ( rank `  x )  C_  B )
6 iunss 4318 . . 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 8112 . . . 4  |-  ( rank `  A )  e.  On
98onssi 6557 . . 3  |-  ( rank `  A )  C_  On
10 eloni 4836 . . 3  |-  ( B  e.  On  ->  Ord  B )
11 ordunisssuc 4928 . . 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 1758   A.wral 2798   _Vcvv 3076    C_ wss 3435   U.cuni 4198   U_ciun 4278   Ord word 4825   Oncon0 4826   suc csuc 4828   ` cfv 5525   rankcrnk 8080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-reg 7917  ax-inf2 7957
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-om 6586  df-recs 6941  df-rdg 6975  df-r1 8081  df-rank 8082
This theorem is referenced by: (None)
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