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Theorem caun0 23994

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 12447	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4238	. . 3 ⊢ ℝ⁺ ≠ ∅
3		iscau2 23990	. . . 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 503	. . 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 4391	. . 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 590	. 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 12310	. . . . . 6 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
8		ne0i 4235	. . . . . 6 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑗) ≠ ∅)
9		r19.2z 4391	. . . . . . 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 416	. . . . . 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 4235	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ 𝑋 → 𝑋 ≠ ∅)
13	12	3ad2ant2 1131	. . . . . 6 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
14	13	rexlimivw 3206	. . . . 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 3204	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → 𝑋 ≠ ∅)
17	16	rexlimivw 3206	. 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 399 ∧ w3a 1084 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 ∃wrex 3071 ∅c0 4227 class class class wbr 5036 dom cdm 5528 ‘cfv 6340 (class class class)co 7156 ↑_pm cpm 8423 ℂcc 10586 1c1 10589 < clt 10726 ℤcz 12033 ℤ_≥cuz 12295 ℝ⁺crp 12443 ∞Metcxmet 20164 Cauccau 23966

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 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-po 5447 df-so 5448 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7699 df-2nd 7700 df-er 8305 df-map 8424 df-pm 8425 df-en 8541 df-dom 8542 df-sdom 8543 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-z 12034 df-uz 12296 df-rp 12444 df-xadd 12562 df-psmet 20171 df-xmet 20172 df-bl 20174 df-cau 23969

This theorem is referenced by: cmetcau 24002

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