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

Theorem infcntss 7316
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 7057 . 2  |-  ( om  ~<_  A  <->  E. x ( om 
~~  x  /\  x  C_  A ) )
3 ensym 7092 . . . . 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 1717   _Vcvv 2899    C_ wss 3263   class class class wbr 4153   omcom 4785    ~~ cen 7042    ~<_ cdom 7043
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-er 6841  df-en 7046  df-dom 7047
  Copyright terms: Public domain W3C validator