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Theorem rankr1a 8055
Description: A relationship between rank and  R1, clearly equivalent to ssrankr1 8054 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8083 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
Hypothesis
Ref Expression
rankid.1  |-  A  e. 
_V
Assertion
Ref Expression
rankr1a  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ( rank `  A )  e.  B
) )

Proof of Theorem rankr1a
StepHypRef Expression
1 rankid.1 . . . 4  |-  A  e. 
_V
21ssrankr1 8054 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  A  e.  ( R1 `  B ) ) )
3 rankon 8014 . . . 4  |-  ( rank `  A )  e.  On
4 ontri1 4765 . . . 4  |-  ( ( B  e.  On  /\  ( rank `  A )  e.  On )  ->  ( B  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  B
) )
53, 4mpan2 671 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  B
) )
62, 5bitr3d 255 . 2  |-  ( B  e.  On  ->  ( -.  A  e.  ( R1 `  B )  <->  -.  ( rank `  A )  e.  B ) )
76con4bid 293 1  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ( rank `  A )  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    e. wcel 1756   _Vcvv 2984    C_ wss 3340   Oncon0 4731   ` cfv 5430   R1cr1 7981   rankcrnk 7982
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 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-reg 7819  ax-inf2 7859
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-recs 6844  df-rdg 6878  df-r1 7983  df-rank 7984
This theorem is referenced by:  r1val2  8056  r1pwOLD  8065  elhf2  28225
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