MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infcntss Structured version   Unicode version

Theorem infcntss 7851
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 7590 . 2  |-  ( om  ~<_  A  <->  E. x ( om 
~~  x  /\  x  C_  A ) )
3 ensym 7625 . . . . 5  |-  ( om 
~~  x  ->  x  ~~  om )
43anim2i 571 . . . 4  |-  ( ( x  C_  A  /\  om 
~~  x )  -> 
( x  C_  A  /\  x  ~~  om )
)
54ancoms 454 . . 3  |-  ( ( om  ~~  x  /\  x  C_  A )  -> 
( x  C_  A  /\  x  ~~  om )
)
65eximi 1703 . 2  |-  ( E. x ( om  ~~  x  /\  x  C_  A
)  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
72, 6sylbi 198 1  |-  ( om  ~<_  A  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   E.wex 1659    e. wcel 1870   _Vcvv 3087    C_ wss 3442   class class class wbr 4426   omcom 6706    ~~ cen 7574    ~<_ cdom 7575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-er 7371  df-en 7578  df-dom 7579
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator