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Theorem onsdom 7914
Description: Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
onsdom  |-  ( A  e.  dom  card  ->  E. x  e.  On  A  ~<  x )
Distinct variable group:    x, A

Proof of Theorem onsdom
StepHypRef Expression
1 harcl 7558 . 2  |-  (har `  A )  e.  On
2 harsdom 7913 . 2  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
3 breq2 4241 . . 3  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
43rspcev 3058 . 2  |-  ( ( (har `  A )  e.  On  /\  A  ~<  (har
`  A ) )  ->  E. x  e.  On  A  ~<  x )
51, 2, 4sylancr 646 1  |-  ( A  e.  dom  card  ->  E. x  e.  On  A  ~<  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1727   E.wrex 2712   class class class wbr 4237   Oncon0 4610   dom cdm 4907   ` cfv 5483    ~< csdm 7137  harchar 7553   cardccrd 7853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-riota 6578  df-recs 6662  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-oi 7508  df-har 7555  df-card 7857
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