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Theorem metn0 24255

Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

**Assertion**
Ref	Expression
metn0	⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))

**Proof of Theorem metn0**
Step	Hyp	Ref	Expression
1		metf 24225	. . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
2		frel 6696	. . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷)
3		reldm0 5894	. . . . 5 ⊢ (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
5	1	fdmd 6701	. . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
6	5	eqeq1d 2732	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
7	4, 6	bitrd 279	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
8		xpeq0 6136	. . . 4 ⊢ ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅))
9		oridm 904	. . . 4 ⊢ ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅)
10	8, 9	bitri 275	. . 3 ⊢ ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅)
11	7, 10	bitrdi 287	. 2 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅))
12	11	necon3bid 2970	1 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∅c0 4299 × cxp 5639 dom cdm 5641 Rel wrel 5646 ⟶wf 6510 ‘cfv 6514 ℝcr 11074 Metcmet 21257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-map 8804 df-met 21265

This theorem is referenced by: (None)

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