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

Theorem limenpsi 7765
Description: A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypothesis
Ref Expression
limenpsi.1  |-  Lim  A
Assertion
Ref Expression
limenpsi  |-  ( A  e.  V  ->  A  ~~  ( A  \  { (/)
} ) )

Proof of Theorem limenpsi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difexg 4545 . . 3  |-  ( A  e.  V  ->  ( A  \  { (/) } )  e.  _V )
2 limenpsi.1 . . . . . . . 8  |-  Lim  A
3 limsuc 6695 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  <->  suc  x  e.  A
) )
42, 3ax-mp 5 . . . . . . 7  |-  ( x  e.  A  <->  suc  x  e.  A )
54biimpi 199 . . . . . 6  |-  ( x  e.  A  ->  suc  x  e.  A )
6 nsuceq0 5510 . . . . . 6  |-  suc  x  =/=  (/)
75, 6jctir 547 . . . . 5  |-  ( x  e.  A  ->  ( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
8 eldifsn 4088 . . . . 5  |-  ( suc  x  e.  ( A 
\  { (/) } )  <-> 
( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
97, 8sylibr 217 . . . 4  |-  ( x  e.  A  ->  suc  x  e.  ( A  \  { (/) } ) )
10 limord 5489 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
112, 10ax-mp 5 . . . . . 6  |-  Ord  A
12 ordelon 5454 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
1311, 12mpan 684 . . . . 5  |-  ( x  e.  A  ->  x  e.  On )
14 ordelon 5454 . . . . . 6  |-  ( ( Ord  A  /\  y  e.  A )  ->  y  e.  On )
1511, 14mpan 684 . . . . 5  |-  ( y  e.  A  ->  y  e.  On )
16 suc11 5533 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( suc  x  =  suc  y  <->  x  =  y ) )
1713, 15, 16syl2an 485 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( suc  x  =  suc  y  <->  x  =  y ) )
189, 17dom3 7631 . . 3  |-  ( ( A  e.  V  /\  ( A  \  { (/) } )  e.  _V )  ->  A  ~<_  ( A  \  { (/) } ) )
191, 18mpdan 681 . 2  |-  ( A  e.  V  ->  A  ~<_  ( A  \  { (/) } ) )
20 difss 3549 . . 3  |-  ( A 
\  { (/) } ) 
C_  A
21 ssdomg 7633 . . 3  |-  ( A  e.  V  ->  (
( A  \  { (/)
} )  C_  A  ->  ( A  \  { (/)
} )  ~<_  A ) )
2220, 21mpi 20 . 2  |-  ( A  e.  V  ->  ( A  \  { (/) } )  ~<_  A )
23 sbth 7710 . 2  |-  ( ( A  ~<_  ( A  \  { (/) } )  /\  ( A  \  { (/) } )  ~<_  A )  ->  A  ~~  ( A  \  { (/) } ) )
2419, 22, 23syl2anc 673 1  |-  ( A  e.  V  ->  A  ~~  ( A  \  { (/)
} ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959   class class class wbr 4395   Ord word 5429   Oncon0 5430   Lim wlim 5431   suc csuc 5432    ~~ cen 7584    ~<_ cdom 7585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-en 7588  df-dom 7589
This theorem is referenced by:  limensuci  7766  omenps  8178  infdifsn  8180  ominf4  8760
  Copyright terms: Public domain W3C validator