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Theorem infpssr 8719
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 3836 . . 3  |-  ( X 
C.  A  ->  E. y
( y  e.  A  /\  -.  y  e.  X
) )
21adantr 463 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. y ( y  e.  A  /\  -.  y  e.  X ) )
3 eldif 3423 . . . 4  |-  ( y  e.  ( A  \  X )  <->  ( y  e.  A  /\  -.  y  e.  X ) )
4 pssss 3537 . . . . . 6  |-  ( X 
C.  A  ->  X  C_  A )
5 bren 7562 . . . . . . . 8  |-  ( X 
~~  A  <->  E. f 
f : X -1-1-onto-> A )
6 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  f : X -1-1-onto-> A )
7 f1ofo 5805 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X -onto-> A )
8 forn 5780 . . . . . . . . . . . . 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 3061 . . . . . . . . . . . . 13  |-  f  e. 
_V
1110rnex 6717 . . . . . . . . . . . 12  |-  ran  f  e.  _V
129, 11syl6eqelr 2499 . . . . . . . . . . 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 752 . . . . . . . . . . . 12  |-  ( ( ( y  e.  ( A  \  X )  /\  X  C_  A
)  /\  f : X
-1-1-onto-> A )  ->  y  e.  ( A  \  X
) )
15 eqid 2402 . . . . . . . . . . . 12  |-  ( rec ( `' f ,  y )  |`  om )  =  ( rec ( `' f ,  y )  |`  om )
1613, 6, 14, 15infpssrlem5 8718 . . . . . . . . . . 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 432 . . . . . . . . 9  |-  ( ( y  e.  ( A 
\  X )  /\  X  C_  A )  -> 
( f : X -1-1-onto-> A  ->  om  ~<_  A ) )
1918exlimdv 1745 . . . . . . . 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 432 . . . . . 6  |-  ( y  e.  ( A  \  X )  ->  ( X  C_  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
224, 21syl5 30 . . . . 5  |-  ( y  e.  ( A  \  X )  ->  ( X  C.  A  ->  ( X  ~~  A  ->  om  ~<_  A ) ) )
2322impd 429 . . . 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 1743 . 2  |-  ( E. y ( y  e.  A  /\  -.  y  e.  X )  ->  (
( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
)
262, 25mpcom 34 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    \ cdif 3410    C_ wss 3413    C. wpss 3414   class class class wbr 4394   `'ccnv 4821   ran crn 4823    |` cres 4824   -onto->wfo 5566   -1-1-onto->wf1o 5567   omcom 6682   reccrdg 7111    ~~ cen 7550    ~<_ cdom 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-en 7554  df-dom 7555
This theorem is referenced by:  isfin4-2  8725
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