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Theorem infcntss 8776
 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 8505 . 2 (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥𝑥𝐴))
3 ensym 8541 . . . 4 (ω ≈ 𝑥𝑥 ≈ ω)
43anim1ci 618 . . 3 ((ω ≈ 𝑥𝑥𝐴) → (𝑥𝐴𝑥 ≈ ω))
54eximi 1836 . 2 (∃𝑥(ω ≈ 𝑥𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
62, 5sylbi 220 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2115  Vcvv 3479   ⊆ wss 3918   class class class wbr 5047  ωcom 7563   ≈ cen 8489   ≼ cdom 8490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-er 8272  df-en 8493  df-dom 8494 This theorem is referenced by:  pibt2  34736
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