MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  harndom Structured version   Unicode version

Theorem harndom 7866
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 7863 . . 3  |-  (har `  X )  e.  On
21onirri 4909 . 2  |-  -.  (har `  X )  e.  (har
`  X )
3 elharval 7865 . . 3  |-  ( (har
`  X )  e.  (har `  X )  <->  ( (har `  X )  e.  On  /\  (har `  X )  ~<_  X ) )
41, 3mpbiran 909 . 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 1757   class class class wbr 4376   Oncon0 4803   ` cfv 5502    ~<_ cdom 7394  harchar 7858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-recs 6918  df-en 7397  df-dom 7398  df-oi 7811  df-har 7860
This theorem is referenced by:  harcard  8235  harsdom  8252  gchhar  8933  ttac  29509  isnumbasgrplem2  29584
  Copyright terms: Public domain W3C validator