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Theorem harndom 7524
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 7521 . . 3  |-  (har `  X )  e.  On
21onirri 4680 . 2  |-  -.  (har `  X )  e.  (har
`  X )
3 elharval 7523 . . 3  |-  ( (har
`  X )  e.  (har `  X )  <->  ( (har `  X )  e.  On  /\  (har `  X )  ~<_  X ) )
41, 3mpbiran 885 . 2  |-  ( (har
`  X )  e.  (har `  X )  <->  (har
`  X )  ~<_  X )
52, 4mtbi 290 1  |-  -.  (har `  X )  ~<_  X
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1725   class class class wbr 4204   Oncon0 4573   ` cfv 5446    ~<_ cdom 7099  harchar 7516
This theorem is referenced by:  harcard  7857  harsdom  7874  gchhar  8538  ttac  27088  isnumbasgrplem2  27227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-en 7102  df-dom 7103  df-oi 7471  df-har 7518
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