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Theorem caun0 23079

Description: A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

**Assertion**
Ref	Expression
caun0	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅)

Proof of Theorem caun0 Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 11836	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 3923	. . 3 ⊢ ℝ⁺ ≠ ∅
3		iscau2 23075	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
4	3	simplbda 654	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
5		r19.2z 4060	. . 3 ⊢ ((ℝ⁺ ≠ ∅ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → ∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
6	2, 4, 5	sylancr 695	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
7		uzid 11702	. . . . . 6 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
8		ne0i 3921	. . . . . 6 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑗) ≠ ∅)
9		r19.2z 4060	. . . . . . 7 ⊢ (((ℤ_≥‘𝑗) ≠ ∅ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → ∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
10	9	ex 450	. . . . . 6 ⊢ ((ℤ_≥‘𝑗) ≠ ∅ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
11	7, 8, 10	3syl 18	. . . . 5 ⊢ (𝑗 ∈ ℤ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
12		ne0i 3921	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ 𝑋 → 𝑋 ≠ ∅)
13	12	3ad2ant2 1083	. . . . . 6 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
14	13	rexlimivw 3029	. . . . 5 ⊢ (∃𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
15	11, 14	syl6 35	. . . 4 ⊢ (𝑗 ∈ ℤ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅))
16	15	rexlimiv 3027	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
17	16	rexlimivw 3029	. 2 ⊢ (∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
18	6, 17	syl 17	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∅c0 3915 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ↑_pm cpm 7858 ℂcc 9934 1c1 9937 < clt 10074 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 ∞Metcxmt 19731 Caucca 23051

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-psmet 19738 df-xmet 19739 df-bl 19741 df-cau 23054

This theorem is referenced by: cmetcau 23087

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