Theorem infcntss 3443

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.)

Hypothesis

Ref	Expression
infcntss.1	⊢ A ∈ V

Assertion

Ref	Expression
infcntss	⊢ (ω ≼ A → ∃x(x ⊆ A ∧ x ≈ ω))

Distinct variable group(s):   x,A

Proof of Theorem infcntss

Step	Hyp	Ref	Expression
1		infcntss.1	. . 3 ⊢ A ∈ V
2	1	domen 3284	. 2 ⊢ (ω ≼ A ↔ ∃x(ω ≈ x ∧ x ⊆ A))
3		visset 1350	. . . . . 6 ⊢ x ∈ V
4	3	ensym 3317	. . . . 5 ⊢ (ω ≈ x → x ≈ ω)
5	4	anim2i 270	. . . 4 ⊢ ((x ⊆ A ∧ ω ≈ x) → (x ⊆ A ∧ x ≈ ω))
6	5	ancoms 334	. . 3 ⊢ ((ω ≈ x ∧ x ⊆ A) → (x ⊆ A ∧ x ≈ ω))
7	6	19.22i 723	. 2 ⊢ (∃x(ω ≈ x ∧ x ⊆ A) → ∃x(x ⊆ A ∧ x ≈ ω))
8	2, 7	sylbi 174	1 ⊢ (ω ≼ A → ∃x(x ⊆ A ∧ x ≈ ω))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   class class class wbr 2054  ωcom 2372   ≈ cen 3271   ≼ cdom 3272

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-er 3200  df-en 3274  df-dom 3275

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