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

Theorem infcntss 7794
 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
Assertion
Ref Expression
infcntss
Distinct variable group:   ,

Proof of Theorem infcntss
StepHypRef Expression
1 infcntss.1 . . 3
21domen 7529 . 2
3 ensym 7564 . . . . 5
43anim2i 569 . . . 4
54ancoms 453 . . 3
65eximi 1635 . 2
72, 6sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369  wex 1596   wcel 1767  cvv 3113   wss 3476   class class class wbr 4447  com 6684   cen 7513   cdom 7514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-er 7311  df-en 7517  df-dom 7518 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator