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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 23855 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) | |
2 | fveq2 6907 | . . . . . 6 ⊢ (𝐵 = ∅ → (fBas‘𝐵) = (fBas‘∅)) | |
3 | 2 | eleq2d 2825 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) ↔ 𝐹 ∈ (fBas‘∅))) |
4 | 3 | biimpd 229 | . . . 4 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) → 𝐹 ∈ (fBas‘∅))) |
5 | fbasne0 23854 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘∅) → 𝐹 ≠ ∅) | |
6 | n0 4359 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
7 | 5, 6 | sylib 218 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘∅) → ∃𝑥 𝑥 ∈ 𝐹) |
8 | fbelss 23857 | . . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ∅) | |
9 | ss0 4408 | . . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 = ∅) |
11 | simpr 484 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
12 | 10, 11 | eqeltrrd 2840 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → ∅ ∈ 𝐹) |
13 | 7, 12 | exlimddv 1933 | . . . 4 ⊢ (𝐹 ∈ (fBas‘∅) → ∅ ∈ 𝐹) |
14 | 4, 13 | syl6com 37 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐵 = ∅ → ∅ ∈ 𝐹)) |
15 | 14 | necon3bd 2952 | . 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 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 fBascfbas 21370 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-fbas 21379 |
This theorem is referenced by: (None) |
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