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

Theorem infcntss 7126
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 6871 . 2  |-  ( om  ~<_  A  <->  E. x ( om 
~~  x  /\  x  C_  A ) )
3 ensym 6906 . . . . 5  |-  ( om 
~~  x  ->  x  ~~  om )
43anim2i 552 . . . 4  |-  ( ( x  C_  A  /\  om 
~~  x )  -> 
( x  C_  A  /\  x  ~~  om )
)
54ancoms 439 . . 3  |-  ( ( om  ~~  x  /\  x  C_  A )  -> 
( x  C_  A  /\  x  ~~  om )
)
65eximi 1563 . 2  |-  ( E. x ( om  ~~  x  /\  x  C_  A
)  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
72, 6sylbi 187 1  |-  ( om  ~<_  A  ->  E. x
( x  C_  A  /\  x  ~~  om )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1685   _Vcvv 2789    C_ wss 3153   class class class wbr 4024   omcom 4655    ~~ cen 6856    ~<_ cdom 6857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-er 6656  df-en 6860  df-dom 6861
  Copyright terms: Public domain W3C validator