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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 21635 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) | |
2 | fveq2 6191 | . . . . . 6 ⊢ (𝐵 = ∅ → (fBas‘𝐵) = (fBas‘∅)) | |
3 | 2 | eleq2d 2687 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) ↔ 𝐹 ∈ (fBas‘∅))) |
4 | 3 | biimpd 219 | . . . 4 ⊢ (𝐵 = ∅ → (𝐹 ∈ (fBas‘𝐵) → 𝐹 ∈ (fBas‘∅))) |
5 | fbasne0 21634 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘∅) → 𝐹 ≠ ∅) | |
6 | n0 3931 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
7 | 5, 6 | sylib 208 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘∅) → ∃𝑥 𝑥 ∈ 𝐹) |
8 | fbelss 21637 | . . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ∅) | |
9 | ss0 3974 | . . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 = ∅) |
11 | simpr 477 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹) | |
12 | 10, 11 | eqeltrrd 2702 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘∅) ∧ 𝑥 ∈ 𝐹) → ∅ ∈ 𝐹) |
13 | 7, 12 | exlimddv 1863 | . . . 4 ⊢ (𝐹 ∈ (fBas‘∅) → ∅ ∈ 𝐹) |
14 | 4, 13 | syl6com 37 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐵 = ∅ → ∅ ∈ 𝐹)) |
15 | 14 | necon3bd 2808 | . 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 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 ‘cfv 5888 fBascfbas 19734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-fbas 19743 |
This theorem is referenced by: (None) |
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