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Theorem limenpsi 7282
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 4351 . . 3  |-  ( A  e.  V  ->  ( A  \  { (/) } )  e.  _V )
2 limenpsi.1 . . . . . . . 8  |-  Lim  A
3 limsuc 4829 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  <->  suc  x  e.  A
) )
42, 3ax-mp 8 . . . . . . 7  |-  ( x  e.  A  <->  suc  x  e.  A )
54biimpi 187 . . . . . 6  |-  ( x  e.  A  ->  suc  x  e.  A )
6 nsuceq0 4661 . . . . . 6  |-  suc  x  =/=  (/)
75, 6jctir 525 . . . . 5  |-  ( x  e.  A  ->  ( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
8 eldifsn 3927 . . . . 5  |-  ( suc  x  e.  ( A 
\  { (/) } )  <-> 
( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
97, 8sylibr 204 . . . 4  |-  ( x  e.  A  ->  suc  x  e.  ( A  \  { (/) } ) )
10 limord 4640 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
112, 10ax-mp 8 . . . . . 6  |-  Ord  A
12 ordelon 4605 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
1311, 12mpan 652 . . . . 5  |-  ( x  e.  A  ->  x  e.  On )
14 ordelon 4605 . . . . . 6  |-  ( ( Ord  A  /\  y  e.  A )  ->  y  e.  On )
1511, 14mpan 652 . . . . 5  |-  ( y  e.  A  ->  y  e.  On )
16 suc11 4685 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( suc  x  =  suc  y  <->  x  =  y ) )
1713, 15, 16syl2an 464 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( suc  x  =  suc  y  <->  x  =  y ) )
189, 17dom3 7151 . . 3  |-  ( ( A  e.  V  /\  ( A  \  { (/) } )  e.  _V )  ->  A  ~<_  ( A  \  { (/) } ) )
191, 18mpdan 650 . 2  |-  ( A  e.  V  ->  A  ~<_  ( A  \  { (/) } ) )
20 difss 3474 . . 3  |-  ( A 
\  { (/) } ) 
C_  A
21 ssdomg 7153 . . 3  |-  ( A  e.  V  ->  (
( A  \  { (/)
} )  C_  A  ->  ( A  \  { (/)
} )  ~<_  A ) )
2220, 21mpi 17 . 2  |-  ( A  e.  V  ->  ( A  \  { (/) } )  ~<_  A )
23 sbth 7227 . 2  |-  ( ( A  ~<_  ( A  \  { (/) } )  /\  ( A  \  { (/) } )  ~<_  A )  ->  A  ~~  ( A  \  { (/) } ) )
2419, 22, 23syl2anc 643 1  |-  ( A  e.  V  ->  A  ~~  ( A  \  { (/)
} ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   {csn 3814   class class class wbr 4212   Ord word 4580   Oncon0 4581   Lim wlim 4582   suc csuc 4583    ~~ cen 7106    ~<_ cdom 7107
This theorem is referenced by:  limensuci  7283  omenps  7609  infdifsn  7611  ominf4  8192
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-en 7110  df-dom 7111
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