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Theorem infpssr 8189
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 3694 . . 3  |-  ( X 
C.  A  ->  E. y
( y  e.  A  /\  -.  y  e.  X
) )
21adantr 453 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. y ( y  e.  A  /\  -.  y  e.  X ) )
3 eldif 3331 . . . 4  |-  ( y  e.  ( A  \  X )  <->  ( y  e.  A  /\  -.  y  e.  X ) )
4 pssss 3443 . . . . . 6  |-  ( X 
C.  A  ->  X  C_  A )
5 bren 7118 . . . . . . . 8  |-  ( X 
~~  A  <->  E. f 
f : X -1-1-onto-> A )
6 simpr 449 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  f : X -1-1-onto-> A )
7 f1ofo 5682 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X -onto-> A )
8 forn 5657 . . . . . . . . . . . . 13  |-  ( f : X -onto-> A  ->  ran  f  =  A
)
96, 7, 83syl 19 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  ran  f  =  A )
10 vex 2960 . . . . . . . . . . . . 13  |-  f  e. 
_V
1110rnex 5134 . . . . . . . . . . . 12  |-  ran  f  e.  _V
129, 11syl6eqelr 2526 . . . . . . . . . . 11  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  A  e.  _V )
13 simplr 733 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  X  C_  A )
14 simpll 732 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  y  e.  ( A  \  X
) )
15 eqid 2437 . . . . . . . . . . . 12  |-  ( rec ( `' f ,  y )  |`  om )  =  ( rec ( `' f ,  y )  |`  om )
1613, 6, 14, 15infpssrlem5 8188 . . . . . . . . . . 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 425 . . . . . . . . 9  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( f : X -1-1-onto-> A  ->  om  ~<_  A ) )
1918exlimdv 1647 . . . . . . . 8  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( E. f  f : X -1-1-onto-> A  ->  om  ~<_  A ) )
205, 19syl5bi 210 . . . . . . 7  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( X  ~~  A  ->  om  ~<_  A ) )
2120ex 425 . . . . . 6  |-  ( y  e.  ( A  \  X )  ->  ( X  C_  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
224, 21syl5 31 . . . . 5  |-  ( y  e.  ( A  \  X )  ->  ( X  C.  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
2322imp3a 422 . . . 4  |-  ( y  e.  ( A  \  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
243, 23sylbir 206 . . 3  |-  ( ( y  e.  A  /\  -.  y  e.  X
)  ->  ( ( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A ) )
2524exlimiv 1645 . 2  |-  ( E. y ( y  e.  A  /\  -.  y  e.  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
262, 25mpcom 35 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om 
~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2957    \ cdif 3318    C_ wss 3321    C. wpss 3322   class class class wbr 4213   omcom 4846   `'ccnv 4878   ran crn 4880    |` cres 4881   -onto->wfo 5453   -1-1-onto->wf1o 5454   reccrdg 6668    ~~ cen 7107    ~<_ cdom 7108
This theorem is referenced by:  isfin4-2  8195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-recs 6634  df-rdg 6669  df-en 7111  df-dom 7112
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