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Theorem infcntss 7583
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 7321 . 2  |-  ( om  ~<_  A  <->  E. x ( om 
~~  x  /\  x  C_  A ) )
3 ensym 7356 . . . . 5  |-  ( om 
~~  x  ->  x  ~~  om )
43anim2i 569 . . . 4  |-  ( ( x  C_  A  /\  om 
~~  x )  -> 
( x  C_  A  /\  x  ~~  om )
)
54ancoms 453 . . 3  |-  ( ( om  ~~  x  /\  x  C_  A )  -> 
( x  C_  A  /\  x  ~~  om )
)
65eximi 1625 . 2  |-  ( E. x ( om  ~~  x  /\  x  C_  A
)  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
72, 6sylbi 195 1  |-  ( om  ~<_  A  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1586    e. wcel 1756   _Vcvv 2970    C_ wss 3326   class class class wbr 4290   omcom 6474    ~~ cen 7305    ~<_ cdom 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-er 7099  df-en 7309  df-dom 7310
This theorem is referenced by: (None)
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