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Theorem infpssr 8738
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 3832 . . 3  |-  ( X 
C.  A  ->  E. y
( y  e.  A  /\  -.  y  e.  X
) )
21adantr 467 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. y ( y  e.  A  /\  -.  y  e.  X ) )
3 eldif 3414 . . . 4  |-  ( y  e.  ( A  \  X )  <->  ( y  e.  A  /\  -.  y  e.  X ) )
4 pssss 3528 . . . . . 6  |-  ( X 
C.  A  ->  X  C_  A )
5 bren 7578 . . . . . . . 8  |-  ( X 
~~  A  <->  E. f 
f : X -1-1-onto-> A )
6 simpr 463 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  f : X -1-1-onto-> A )
7 f1ofo 5821 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X -onto-> A )
8 forn 5796 . . . . . . . . . . . . 13  |-  ( f : X -onto-> A  ->  ran  f  =  A
)
96, 7, 83syl 18 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  ran  f  =  A )
10 vex 3048 . . . . . . . . . . . . 13  |-  f  e. 
_V
1110rnex 6727 . . . . . . . . . . . 12  |-  ran  f  e.  _V
129, 11syl6eqelr 2538 . . . . . . . . . . 11  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  A  e.  _V )
13 simplr 762 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  X  C_  A )
14 simpll 760 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  y  e.  ( A  \  X
) )
15 eqid 2451 . . . . . . . . . . . 12  |-  ( rec ( `' f ,  y )  |`  om )  =  ( rec ( `' f ,  y )  |`  om )
1613, 6, 14, 15infpssrlem5 8737 . . . . . . . . . . 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 436 . . . . . . . . 9  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( f : X -1-1-onto-> A  ->  om  ~<_  A ) )
1918exlimdv 1779 . . . . . . . 8  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( E. f  f : X -1-1-onto-> A  ->  om  ~<_  A ) )
205, 19syl5bi 221 . . . . . . 7  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( X  ~~  A  ->  om  ~<_  A ) )
2120ex 436 . . . . . 6  |-  ( y  e.  ( A  \  X )  ->  ( X  C_  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
224, 21syl5 33 . . . . 5  |-  ( y  e.  ( A  \  X )  ->  ( X  C.  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
2322impd 433 . . . 4  |-  ( y  e.  ( A  \  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
243, 23sylbir 217 . . 3  |-  ( ( y  e.  A  /\  -.  y  e.  X
)  ->  ( ( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A ) )
2524exlimiv 1776 . 2  |-  ( E. y ( y  e.  A  /\  -.  y  e.  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
262, 25mpcom 37 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    \ cdif 3401    C_ wss 3404    C. wpss 3405   class class class wbr 4402   `'ccnv 4833   ran crn 4835    |` cres 4836   -onto->wfo 5580   -1-1-onto->wf1o 5581   omcom 6692   reccrdg 7127    ~~ cen 7566    ~<_ cdom 7567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-en 7570  df-dom 7571
This theorem is referenced by:  isfin4-2  8744
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