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Mirrors > Home > MPE Home > Th. List > fbdmn0 | Structured version Visualization version GIF version |
Description: The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
fbdmn0 | ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelfb 22433 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) | |
2 | fveq2 6665 | . . . . . 6 ⊢ (𝐵 = ∅ → (fBas‘𝐵) = (fBas‘∅)) | |
3 | 2 | eleq2d 2898 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) ↔ 𝐹 ∈ (fBas‘∅))) |
4 | 3 | biimpd 231 | . . . 4 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) → 𝐹 ∈ (fBas‘∅))) |
5 | fbasne0 22432 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘∅) → 𝐹 ≠ ∅) | |
6 | n0 4310 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
7 | 5, 6 | sylib 220 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘∅) → ∃𝑥 𝑥 ∈ 𝐹) |
8 | fbelss 22435 | . . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ∅) | |
9 | ss0 4352 | . . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 = ∅) |
11 | simpr 487 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
12 | 10, 11 | eqeltrrd 2914 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → ∅ ∈ 𝐹) |
13 | 7, 12 | exlimddv 1932 | . . . 4 ⊢ (𝐹 ∈ (fBas‘∅) → ∅ ∈ 𝐹) |
14 | 4, 13 | syl6com 37 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐵 = ∅ → ∅ ∈ 𝐹)) |
15 | 14 | necon3bd 3030 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → (¬ ∅ ∈ 𝐹 → 𝐵 ≠ ∅)) |
16 | 1, 15 | mpd 15 | 1 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 ‘cfv 6350 fBascfbas 20527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fv 6358 df-fbas 20536 |
This theorem is referenced by: (None) |
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