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(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 8505	. 2 ⊢ (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 8541	. . . 4 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω)
4	3	anim1ci 618	. . 3 ⊢ ((ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
5	4	eximi 1836	. 2 ⊢ (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
6	2, 5	sylbi 220	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030  ωcom 7560   ≈ 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-er 8272  df-en 8493  df-dom 8494

This theorem is referenced by:  pibt2  34834