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

Theorem infcntss 8522
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 ∈ V
Assertion
Ref Expression
infcntss (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
Distinct variable group:   𝑥,𝐴

Proof of Theorem infcntss
StepHypRef Expression
1 infcntss.1 . . 3 𝐴 ∈ V
21domen 8254 . 2 (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥𝑥𝐴))
3 ensym 8290 . . . 4 (ω ≈ 𝑥𝑥 ≈ ω)
43anim1ci 609 . . 3 ((ω ≈ 𝑥𝑥𝐴) → (𝑥𝐴𝑥 ≈ ω))
54eximi 1878 . 2 (∃𝑥(ω ≈ 𝑥𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
62, 5sylbi 209 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wex 1823  wcel 2107  Vcvv 3398  wss 3792   class class class wbr 4886  ωcom 7343  cen 8238  cdom 8239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-er 8026  df-en 8242  df-dom 8243
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator