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Theorem limenpsi 7694
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 4585 . . 3  |-  ( A  e.  V  ->  ( A  \  { (/) } )  e.  _V )
2 limenpsi.1 . . . . . . . 8  |-  Lim  A
3 limsuc 6669 . . . . . . . 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 4948 . . . . . 6  |-  suc  x  =/=  (/)
75, 6jctir 538 . . . . 5  |-  ( x  e.  A  ->  ( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
8 eldifsn 4140 . . . . 5  |-  ( suc  x  e.  ( A 
\  { (/) } )  <-> 
( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
97, 8sylibr 212 . . . 4  |-  ( x  e.  A  ->  suc  x  e.  ( A  \  { (/) } ) )
10 limord 4927 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
112, 10ax-mp 5 . . . . . 6  |-  Ord  A
12 ordelon 4892 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
1311, 12mpan 670 . . . . 5  |-  ( x  e.  A  ->  x  e.  On )
14 ordelon 4892 . . . . . 6  |-  ( ( Ord  A  /\  y  e.  A )  ->  y  e.  On )
1511, 14mpan 670 . . . . 5  |-  ( y  e.  A  ->  y  e.  On )
16 suc11 4971 . . . . 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 7561 . . 3  |-  ( ( A  e.  V  /\  ( A  \  { (/) } )  e.  _V )  ->  A  ~<_  ( A  \  { (/) } ) )
191, 18mpdan 668 . 2  |-  ( A  e.  V  ->  A  ~<_  ( A  \  { (/) } ) )
20 difss 3616 . . 3  |-  ( A 
\  { (/) } ) 
C_  A
21 ssdomg 7563 . . 3  |-  ( A  e.  V  ->  (
( A  \  { (/)
} )  C_  A  ->  ( A  \  { (/)
} )  ~<_  A ) )
2220, 21mpi 17 . 2  |-  ( A  e.  V  ->  ( A  \  { (/) } )  ~<_  A )
23 sbth 7639 . 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 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    \ cdif 3458    C_ wss 3461   (/)c0 3770   {csn 4014   class class class wbr 4437   Ord word 4867   Oncon0 4868   Lim wlim 4869   suc csuc 4870    ~~ cen 7515    ~<_ cdom 7516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-en 7519  df-dom 7520
This theorem is referenced by:  limensuci  7695  omenps  8074  infdifsn  8076  ominf4  8695
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