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Theorem limenpsi 7586
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 4538 . . 3  |-  ( A  e.  V  ->  ( A  \  { (/) } )  e.  _V )
2 limenpsi.1 . . . . . . . 8  |-  Lim  A
3 limsuc 6560 . . . . . . . 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 4897 . . . . . 6  |-  suc  x  =/=  (/)
75, 6jctir 538 . . . . 5  |-  ( x  e.  A  ->  ( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
8 eldifsn 4098 . . . . 5  |-  ( suc  x  e.  ( A 
\  { (/) } )  <-> 
( suc  x  e.  A  /\  suc  x  =/=  (/) ) )
97, 8sylibr 212 . . . 4  |-  ( x  e.  A  ->  suc  x  e.  ( A  \  { (/) } ) )
10 limord 4876 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
112, 10ax-mp 5 . . . . . 6  |-  Ord  A
12 ordelon 4841 . . . . . 6  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
1311, 12mpan 670 . . . . 5  |-  ( x  e.  A  ->  x  e.  On )
14 ordelon 4841 . . . . . 6  |-  ( ( Ord  A  /\  y  e.  A )  ->  y  e.  On )
1511, 14mpan 670 . . . . 5  |-  ( y  e.  A  ->  y  e.  On )
16 suc11 4920 . . . . 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 7453 . . 3  |-  ( ( A  e.  V  /\  ( A  \  { (/) } )  e.  _V )  ->  A  ~<_  ( A  \  { (/) } ) )
191, 18mpdan 668 . 2  |-  ( A  e.  V  ->  A  ~<_  ( A  \  { (/) } ) )
20 difss 3581 . . 3  |-  ( A 
\  { (/) } ) 
C_  A
21 ssdomg 7455 . . 3  |-  ( A  e.  V  ->  (
( A  \  { (/)
} )  C_  A  ->  ( A  \  { (/)
} )  ~<_  A ) )
2220, 21mpi 17 . 2  |-  ( A  e.  V  ->  ( A  \  { (/) } )  ~<_  A )
23 sbth 7531 . 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 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3068    \ cdif 3423    C_ wss 3426   (/)c0 3735   {csn 3975   class class class wbr 4390   Ord word 4816   Oncon0 4817   Lim wlim 4818   suc csuc 4819    ~~ cen 7407    ~<_ cdom 7408
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-en 7411  df-dom 7412
This theorem is referenced by:  limensuci  7587  omenps  7961  infdifsn  7963  ominf4  8582
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