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Theorem harndom 8032
Description: The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harndom  |-  -.  (har `  X )  ~<_  X

Proof of Theorem harndom
StepHypRef Expression
1 harcl 8029 . . 3  |-  (har `  X )  e.  On
21onirri 5491 . 2  |-  -.  (har `  X )  e.  (har
`  X )
3 elharval 8031 . . 3  |-  ( (har
`  X )  e.  (har `  X )  <->  ( (har `  X )  e.  On  /\  (har `  X )  ~<_  X ) )
41, 3mpbiran 926 . 2  |-  ( (har
`  X )  e.  (har `  X )  <->  (har
`  X )  ~<_  X )
52, 4mtbi 299 1  |-  -.  (har `  X )  ~<_  X
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1872   class class class wbr 4366   Oncon0 5385   ` cfv 5544    ~<_ cdom 7522  harchar 8024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-wrecs 6983  df-recs 7045  df-en 7525  df-dom 7526  df-oi 7978  df-har 8026
This theorem is referenced by:  harcard  8364  harsdom  8381  gchhar  9055  ttac  35804  isnumbasgrplem2  35876
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