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Theorem onsdom 7847
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 7493 . 2  |-  (har `  A )  e.  On
2 harsdom 7846 . 2  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
3 breq2 4184 . . 3  |-  ( x  =  (har `  A
)  ->  ( A  ~<  x  <->  A  ~<  (har `  A ) ) )
43rspcev 3020 . 2  |-  ( ( (har `  A )  e.  On  /\  A  ~<  (har
`  A ) )  ->  E. x  e.  On  A  ~<  x )
51, 2, 4sylancr 645 1  |-  ( A  e.  dom  card  ->  E. x  e.  On  A  ~<  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   E.wrex 2675   class class class wbr 4180   Oncon0 4549   dom cdm 4845   ` cfv 5421    ~< csdm 7075  harchar 7488   cardccrd 7786
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6516  df-recs 6600  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-oi 7443  df-har 7490  df-card 7790
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