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Theorem infcntss 7339
Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
Hypothesis
Ref Expression
infcntss.1  |-  A  e. 
_V
Assertion
Ref Expression
infcntss  |-  ( om  ~<_  A  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
Distinct variable group:    x, A

Proof of Theorem infcntss
StepHypRef Expression
1 infcntss.1 . . 3  |-  A  e. 
_V
21domen 7080 . 2  |-  ( om  ~<_  A  <->  E. x ( om 
~~  x  /\  x  C_  A ) )
3 ensym 7115 . . . . 5  |-  ( om 
~~  x  ->  x  ~~  om )
43anim2i 553 . . . 4  |-  ( ( x  C_  A  /\  om 
~~  x )  -> 
( x  C_  A  /\  x  ~~  om )
)
54ancoms 440 . . 3  |-  ( ( om  ~~  x  /\  x  C_  A )  -> 
( x  C_  A  /\  x  ~~  om )
)
65eximi 1582 . 2  |-  ( E. x ( om  ~~  x  /\  x  C_  A
)  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
72, 6sylbi 188 1  |-  ( om  ~<_  A  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    e. wcel 1721   _Vcvv 2916    C_ wss 3280   class class class wbr 4172   omcom 4804    ~~ cen 7065    ~<_ cdom 7066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-er 6864  df-en 7069  df-dom 7070
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