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Theorem harndom 7988
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 7985 . . 3  |-  (har `  X )  e.  On
21onirri 4970 . 2  |-  -.  (har `  X )  e.  (har
`  X )
3 elharval 7987 . . 3  |-  ( (har
`  X )  e.  (har `  X )  <->  ( (har `  X )  e.  On  /\  (har `  X )  ~<_  X ) )
41, 3mpbiran 916 . 2  |-  ( (har
`  X )  e.  (har `  X )  <->  (har
`  X )  ~<_  X )
52, 4mtbi 298 1  |-  -.  (har `  X )  ~<_  X
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1802   class class class wbr 4433   Oncon0 4864   ` cfv 5574    ~<_ cdom 7512  harchar 7980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-recs 7040  df-en 7515  df-dom 7516  df-oi 7933  df-har 7982
This theorem is referenced by:  harcard  8357  harsdom  8374  gchhar  9055  ttac  30946  isnumbasgrplem2  31021
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