Theorem infcntss 9256

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(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Distinct variable group:   𝑥,𝐴

Proof of Theorem infcntss

Step	Hyp	Ref	Expression
1		infcntss.1	. . 3 ⊢ 𝐴 ∈ V
2	1	domen 8931	. 2 ⊢ (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 8973	. . . 4 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω)
4	3	anim1ci 624	. . 3 ⊢ ((ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
5	4	eximi 1849	. 2 ⊢ (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
6	2, 5	sylbi 219	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1793   ∈ wcel 2136  Vcvv 3448   ⊆ wss 3899   class class class wbr 5094  ωcom 7835   ≈ cen 8913   ≼ cdom 8914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-er 8666  df-en 8917  df-dom 8918

This theorem is referenced by:  pibt2  37859