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Theorem infpssr 7934
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 3519 . . 3  |-  ( X 
C.  A  ->  E. y
( y  e.  A  /\  -.  y  e.  X
) )
21adantr 451 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. y ( y  e.  A  /\  -.  y  e.  X ) )
3 eldif 3162 . . . 4  |-  ( y  e.  ( A  \  X )  <->  ( y  e.  A  /\  -.  y  e.  X ) )
4 pssss 3271 . . . . . 6  |-  ( X 
C.  A  ->  X  C_  A )
5 bren 6871 . . . . . . . 8  |-  ( X 
~~  A  <->  E. f 
f : X -1-1-onto-> A )
6 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  f : X -1-1-onto-> A )
7 f1ofo 5479 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X -onto-> A )
8 forn 5454 . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . 13  |-  f  e. 
_V
1110rnex 4942 . . . . . . . . . . . 12  |-  ran  f  e.  _V
129, 11syl6eqelr 2372 . . . . . . . . . . 11  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  A  e.  _V )
13 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  X  C_  A )
14 simpll 730 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  y  e.  ( A  \  X
) )
15 eqid 2283 . . . . . . . . . . . 12  |-  ( rec ( `' f ,  y )  |`  om )  =  ( rec ( `' f ,  y )  |`  om )
1613, 6, 14, 15infpssrlem5 7933 . . . . . . . . . . 11  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  ( A  e.  _V  ->  om  ~<_  A ) )
1712, 16mpd 14 . . . . . . . . . 10  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  om  ~<_  A )
1817ex 423 . . . . . . . . 9  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( f : X -1-1-onto-> A  ->  om  ~<_  A ) )
1918exlimdv 1664 . . . . . . . 8  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( E. f  f : X -1-1-onto-> A  ->  om  ~<_  A ) )
205, 19syl5bi 208 . . . . . . 7  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( X  ~~  A  ->  om  ~<_  A ) )
2120ex 423 . . . . . 6  |-  ( y  e.  ( A  \  X )  ->  ( X  C_  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
224, 21syl5 28 . . . . 5  |-  ( y  e.  ( A  \  X )  ->  ( X  C.  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
2322imp3a 420 . . . 4  |-  ( y  e.  ( A  \  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
243, 23sylbir 204 . . 3  |-  ( ( y  e.  A  /\  -.  y  e.  X
)  ->  ( ( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A ) )
2524exlimiv 1666 . 2  |-  ( E. y ( y  e.  A  /\  -.  y  e.  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
262, 25mpcom 32 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om 
~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152    C. wpss 3153   class class class wbr 4023   omcom 4656   `'ccnv 4688   ran crn 4690    |` cres 4691   -onto->wfo 5253   -1-1-onto->wf1o 5254   reccrdg 6422    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  isfin4-2  7940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-en 6864  df-dom 6865
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