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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 21445 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝐵 = ∅ → (fBas‘𝐵) = (fBas‘∅)) | |
3 | 2 | eleq2d 2673 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) ↔ 𝐹 ∈ (fBas‘∅))) |
4 | 3 | biimpd 218 | . . . 4 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) → 𝐹 ∈ (fBas‘∅))) |
5 | fbasne0 21444 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘∅) → 𝐹 ≠ ∅) | |
6 | n0 3890 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
7 | 5, 6 | sylib 207 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘∅) → ∃𝑥 𝑥 ∈ 𝐹) |
8 | fbelss 21447 | . . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ∅) | |
9 | ss0 3926 | . . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 = ∅) |
11 | simpr 476 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
12 | 10, 11 | eqeltrrd 2689 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → ∅ ∈ 𝐹) |
13 | 7, 12 | exlimddv 1850 | . . . 4 ⊢ (𝐹 ∈ (fBas‘∅) → ∅ ∈ 𝐹) |
14 | 4, 13 | syl6com 36 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐵 = ∅ → ∅ ∈ 𝐹)) |
15 | 14 | necon3bd 2796 | . 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 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 fBascfbas 19555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-fbas 19564 |
This theorem is referenced by: (None) |
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