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Theorem rankonid 7496
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 7495 . . 3  |-  ( A  e.  dom  R1  ->  ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  A ) )
21simprd 451 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
3 id 21 . . 3  |-  ( (
rank `  A )  =  A  ->  ( rank `  A )  =  A )
4 rankdmr1 7468 . . 3  |-  ( rank `  A )  e.  dom  R1
53, 4syl6eqelr 2373 . 2  |-  ( (
rank `  A )  =  A  ->  A  e. 
dom  R1 )
62, 5impbii 182 1  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1628    e. wcel 1688   U.cuni 3828   Oncon0 4391   dom cdm 4688   "cima 4691   ` cfv 5221   R1cr1 7429   rankcrnk 7430
This theorem is referenced by:  rankeq0b  7527  rankr1id  7529  rankcf  8394  r1tskina  8399  rankeq1o  24208  hfninf  24223
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383  df-rdg 6418  df-r1 7431  df-rank 7432
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