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Theorem rankr1a 7762
Description: A relationship between rank and  R1, clearly equivalent to ssrankr1 7761 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7790 for the subset verion. (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 7761 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  A  e.  ( R1 `  B ) ) )
3 rankon 7721 . . . 4  |-  ( rank `  A )  e.  On
4 ontri1 4615 . . . 4  |-  ( ( B  e.  On  /\  ( rank `  A )  e.  On )  ->  ( B  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  B
) )
53, 4mpan2 653 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  B
) )
62, 5bitr3d 247 . 2  |-  ( B  e.  On  ->  ( -.  A  e.  ( R1 `  B )  <->  -.  ( rank `  A )  e.  B ) )
76con4bid 285 1  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ( rank `  A )  e.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    e. wcel 1725   _Vcvv 2956    C_ wss 3320   Oncon0 4581   ` cfv 5454   R1cr1 7688   rankcrnk 7689
This theorem is referenced by:  r1val2  7763  r1pwOLD  7772  elhf2  26116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690  df-rank 7691
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