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Theorem onwf 7690
Description: The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
onwf  |-  On  C_  U. ( R1 " On )

Proof of Theorem onwf
StepHypRef Expression
1 r1fnon 7627 . . 3  |-  R1  Fn  On
2 fndm 5485 . . 3  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 8 . 2  |-  dom  R1  =  On
4 rankonidlem 7688 . . . 4  |-  ( x  e.  dom  R1  ->  ( x  e.  U. ( R1 " On )  /\  ( rank `  x )  =  x ) )
54simpld 446 . . 3  |-  ( x  e.  dom  R1  ->  x  e.  U. ( R1
" On ) )
65ssriv 3296 . 2  |-  dom  R1  C_ 
U. ( R1 " On )
73, 6eqsstr3i 3323 1  |-  On  C_  U. ( R1 " On )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    C_ wss 3264   U.cuni 3958   Oncon0 4523   dom cdm 4819   "cima 4822    Fn wfn 5390   ` cfv 5395   R1cr1 7622   rankcrnk 7623
This theorem is referenced by:  dfac12r  7960  r1tskina  8591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605  df-r1 7624  df-rank 7625
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