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Theorem harsdom 7884
Description: The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harsdom  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)

Proof of Theorem harsdom
StepHypRef Expression
1 harndom 7534 . 2  |-  -.  (har `  A )  ~<_  A
2 harcl 7531 . . . 4  |-  (har `  A )  e.  On
3 onenon 7838 . . . 4  |-  ( (har
`  A )  e.  On  ->  (har `  A
)  e.  dom  card )
42, 3ax-mp 8 . . 3  |-  (har `  A )  e.  dom  card
5 domtri2 7878 . . . 4  |-  ( ( (har `  A )  e.  dom  card  /\  A  e. 
dom  card )  ->  (
(har `  A )  ~<_  A 
<->  -.  A  ~<  (har `  A ) ) )
65con2bid 321 . . 3  |-  ( ( (har `  A )  e.  dom  card  /\  A  e. 
dom  card )  ->  ( A  ~<  (har `  A
)  <->  -.  (har `  A
)  ~<_  A ) )
74, 6mpan 653 . 2  |-  ( A  e.  dom  card  ->  ( A  ~<  (har `  A
)  <->  -.  (har `  A
)  ~<_  A ) )
81, 7mpbiri 226 1  |-  ( A  e.  dom  card  ->  A 
~<  (har `  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   class class class wbr 4214   Oncon0 4583   dom cdm 4880   ` cfv 5456    ~<_ cdom 7109    ~< csdm 7110  harchar 7526   cardccrd 7824
This theorem is referenced by:  onsdom  7885  harval2  7886  alephordilem1  7956  gchaleph2  8553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-riota 6551  df-recs 6635  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-oi 7481  df-har 7528  df-card 7828
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