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Theorem rankonid 7739
Description: The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankonid  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )

Proof of Theorem rankonid
StepHypRef Expression
1 rankonidlem 7738 . . 3  |-  ( A  e.  dom  R1  ->  ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  A ) )
21simprd 450 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
3 id 20 . . 3  |-  ( (
rank `  A )  =  A  ->  ( rank `  A )  =  A )
4 rankdmr1 7711 . . 3  |-  ( rank `  A )  e.  dom  R1
53, 4syl6eqelr 2519 . 2  |-  ( (
rank `  A )  =  A  ->  A  e. 
dom  R1 )
62, 5impbii 181 1  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   U.cuni 4002   Oncon0 4568   dom cdm 4864   "cima 4867   ` cfv 5440   R1cr1 7672   rankcrnk 7673
This theorem is referenced by:  rankeq0b  7770  rankr1id  7772  rankcf  8636  r1tskina  8641  rankeq1o  26055  hfninf  26070
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
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 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-recs 6619  df-rdg 6654  df-r1 7674  df-rank 7675
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