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

Theorem infpssr 8469
Description: Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infpssr  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om 
~<_  A )

Proof of Theorem infpssr
Dummy variables  y 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pssnel 3739 . . 3  |-  ( X 
C.  A  ->  E. y
( y  e.  A  /\  -.  y  e.  X
) )
21adantr 465 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. y ( y  e.  A  /\  -.  y  e.  X ) )
3 eldif 3333 . . . 4  |-  ( y  e.  ( A  \  X )  <->  ( y  e.  A  /\  -.  y  e.  X ) )
4 pssss 3446 . . . . . 6  |-  ( X 
C.  A  ->  X  C_  A )
5 bren 7311 . . . . . . . 8  |-  ( X 
~~  A  <->  E. f 
f : X -1-1-onto-> A )
6 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  f : X -1-1-onto-> A )
7 f1ofo 5643 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X -onto-> A )
8 forn 5618 . . . . . . . . . . . . 13  |-  ( f : X -onto-> A  ->  ran  f  =  A
)
96, 7, 83syl 20 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  ran  f  =  A )
10 vex 2970 . . . . . . . . . . . . 13  |-  f  e. 
_V
1110rnex 6507 . . . . . . . . . . . 12  |-  ran  f  e.  _V
129, 11syl6eqelr 2527 . . . . . . . . . . 11  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  A  e.  _V )
13 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  X  C_  A )
14 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  y  e.  ( A  \  X
) )
15 eqid 2438 . . . . . . . . . . . 12  |-  ( rec ( `' f ,  y )  |`  om )  =  ( rec ( `' f ,  y )  |`  om )
1613, 6, 14, 15infpssrlem5 8468 . . . . . . . . . . 11  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  ( A  e.  _V  ->  om  ~<_  A ) )
1712, 16mpd 15 . . . . . . . . . 10  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  om  ~<_  A )
1817ex 434 . . . . . . . . 9  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( f : X -1-1-onto-> A  ->  om  ~<_  A ) )
1918exlimdv 1690 . . . . . . . 8  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( E. f  f : X -1-1-onto-> A  ->  om  ~<_  A ) )
205, 19syl5bi 217 . . . . . . 7  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( X  ~~  A  ->  om  ~<_  A ) )
2120ex 434 . . . . . 6  |-  ( y  e.  ( A  \  X )  ->  ( X  C_  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
224, 21syl5 32 . . . . 5  |-  ( y  e.  ( A  \  X )  ->  ( X  C.  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
2322impd 431 . . . 4  |-  ( y  e.  ( A  \  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
243, 23sylbir 213 . . 3  |-  ( ( y  e.  A  /\  -.  y  e.  X
)  ->  ( ( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A ) )
2524exlimiv 1688 . 2  |-  ( E. y ( y  e.  A  /\  -.  y  e.  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
262, 25mpcom 36 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2967    \ cdif 3320    C_ wss 3323    C. wpss 3324   class class class wbr 4287   `'ccnv 4834   ran crn 4836    |` cres 4837   -onto->wfo 5411   -1-1-onto->wf1o 5412   omcom 6471   reccrdg 6857    ~~ 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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-en 7303  df-dom 7304
This theorem is referenced by:  isfin4-2  8475
  Copyright terms: Public domain W3C validator