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Theorem infcntss 9272
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 8908 . 2 (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥𝑥𝐴))
3 ensym 8950 . . . 4 (ω ≈ 𝑥𝑥 ≈ ω)
43anim1ci 617 . . 3 ((ω ≈ 𝑥𝑥𝐴) → (𝑥𝐴𝑥 ≈ ω))
54eximi 1838 . 2 (∃𝑥(ω ≈ 𝑥𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
62, 5sylbi 216 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  Vcvv 3448  wss 3915   class class class wbr 5110  ωcom 7807  cen 8887  cdom 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-er 8655  df-en 8891  df-dom 8892
This theorem is referenced by:  pibt2  35917
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