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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 22436 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) | |
2 | fveq2 6645 | . . . . . 6 ⊢ (𝐵 = ∅ → (fBas‘𝐵) = (fBas‘∅)) | |
3 | 2 | eleq2d 2875 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) ↔ 𝐹 ∈ (fBas‘∅))) |
4 | 3 | biimpd 232 | . . . 4 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) → 𝐹 ∈ (fBas‘∅))) |
5 | fbasne0 22435 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘∅) → 𝐹 ≠ ∅) | |
6 | n0 4260 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
7 | 5, 6 | sylib 221 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘∅) → ∃𝑥 𝑥 ∈ 𝐹) |
8 | fbelss 22438 | . . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ∅) | |
9 | ss0 4306 | . . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 = ∅) |
11 | simpr 488 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
12 | 10, 11 | eqeltrrd 2891 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → ∅ ∈ 𝐹) |
13 | 7, 12 | exlimddv 1936 | . . . 4 ⊢ (𝐹 ∈ (fBas‘∅) → ∅ ∈ 𝐹) |
14 | 4, 13 | syl6com 37 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐵 = ∅ → ∅ ∈ 𝐹)) |
15 | 14 | necon3bd 3001 | . 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 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ‘cfv 6324 fBascfbas 20079 |
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 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-fbas 20088 |
This theorem is referenced by: (None) |
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