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Theorem rankr1a 7553
Description: A relationship between rank and  R1, clearly equivalent to ssrankr1 7552 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7581 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 7552 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  A  e.  ( R1 `  B ) ) )
3 rankon 7512 . . . 4  |-  ( rank `  A )  e.  On
4 ontri1 4463 . . . 4  |-  ( ( B  e.  On  /\  ( rank `  A )  e.  On )  ->  ( B  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  B
) )
53, 4mpan2 652 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  B
) )
62, 5bitr3d 246 . 2  |-  ( B  e.  On  ->  ( -.  A  e.  ( R1 `  B )  <->  -.  ( rank `  A )  e.  B ) )
76con4bid 284 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 176    e. wcel 1701   _Vcvv 2822    C_ wss 3186   Oncon0 4429   ` cfv 5292   R1cr1 7479   rankcrnk 7480
This theorem is referenced by:  r1val2  7554  r1pwOLD  7563  elhf2  25191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-reg 7351  ax-inf2 7387
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430  df-rdg 6465  df-r1 7481  df-rank 7482
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