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Theorem limenpsi 7478
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 4433 . . 3  |-  ( A  e.  V  ->  ( A  \  { (/) } )  e.  _V )
2 limenpsi.1 . . . . . . . 8  |-  Lim  A
3 limsuc 6455 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  <->  suc  x  e.  A
) )
42, 3ax-mp 5 . . . . . . 7  |-  ( x  e.  A  <->  suc  x  e.  A )
54biimpi 194 . . . . . 6  |-  ( x  e.  A  ->  suc  x  e.  A )
6 nsuceq0 4791 . . . . . 6  |-  suc  x  =/=  (/)
75, 6jctir 538 . . . . 5  |-  ( x  e.  A  ->  ( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
8 eldifsn 3993 . . . . 5  |-  ( suc  x  e.  ( A 
\  { (/) } )  <-> 
( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
97, 8sylibr 212 . . . 4  |-  ( x  e.  A  ->  suc  x  e.  ( A  \  { (/) } ) )
10 limord 4770 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
112, 10ax-mp 5 . . . . . 6  |-  Ord  A
12 ordelon 4735 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
1311, 12mpan 670 . . . . 5  |-  ( x  e.  A  ->  x  e.  On )
14 ordelon 4735 . . . . . 6  |-  ( ( Ord  A  /\  y  e.  A )  ->  y  e.  On )
1511, 14mpan 670 . . . . 5  |-  ( y  e.  A  ->  y  e.  On )
16 suc11 4814 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( suc  x  =  suc  y  <->  x  =  y ) )
1713, 15, 16syl2an 477 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( suc  x  =  suc  y  <->  x  =  y ) )
189, 17dom3 7345 . . 3  |-  ( ( A  e.  V  /\  ( A  \  { (/) } )  e.  _V )  ->  A  ~<_  ( A  \  { (/) } ) )
191, 18mpdan 668 . 2  |-  ( A  e.  V  ->  A  ~<_  ( A  \  { (/) } ) )
20 difss 3476 . . 3  |-  ( A 
\  { (/) } ) 
C_  A
21 ssdomg 7347 . . 3  |-  ( A  e.  V  ->  (
( A  \  { (/)
} )  C_  A  ->  ( A  \  { (/)
} )  ~<_  A ) )
2220, 21mpi 17 . 2  |-  ( A  e.  V  ->  ( A  \  { (/) } )  ~<_  A )
23 sbth 7423 . 2  |-  ( ( A  ~<_  ( A  \  { (/) } )  /\  ( A  \  { (/) } )  ~<_  A )  ->  A  ~~  ( A  \  { (/) } ) )
2419, 22, 23syl2anc 661 1  |-  ( A  e.  V  ->  A  ~~  ( A  \  { (/)
} ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2600   _Vcvv 2966    \ cdif 3318    C_ wss 3321   (/)c0 3630   {csn 3870   class class class wbr 4285   Ord word 4710   Oncon0 4711   Lim wlim 4712   suc csuc 4713    ~~ cen 7299    ~<_ cdom 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-en 7303  df-dom 7304
This theorem is referenced by:  limensuci  7479  omenps  7852  infdifsn  7854  ominf4  8473
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