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Theorem rankr1a 8271
Description: A relationship between rank and  R1, clearly equivalent to ssrankr1 8270 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8299 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 8270 . . 3  |-  ( B  e.  On  ->  ( B  C_  ( rank `  A
)  <->  -.  A  e.  ( R1 `  B ) ) )
3 rankon 8230 . . . 4  |-  ( rank `  A )  e.  On
4 ontri1 4921 . . . 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 1819   _Vcvv 3109    C_ wss 3471   Oncon0 4887   ` cfv 5594   R1cr1 8197   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:  r1val2  8272  r1pwALT  8281  elhf2  29994
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