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Mirrors > Home > MPE Home > Th. List > 0nelfb | Structured version Visualization version GIF version |
Description: No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
0nelfb | ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6130 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ∈ dom fBas) | |
2 | isfbas 21443 | . . . . 5 ⊢ (𝐵 ∈ dom fBas → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) |
4 | 3 | ibi 255 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))) |
5 | simpr2 1061 | . . 3 ⊢ ((𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)) → ∅ ∉ 𝐹) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → ∅ ∉ 𝐹) |
7 | df-nel 2783 | . 2 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹) | |
8 | 6, 7 | sylib 207 | 1 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 dom cdm 5038 ‘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: fbdmn0 21448 fbncp 21453 fbun 21454 fbfinnfr 21455 0nelfil 21463 fsubbas 21481 fbasfip 21482 fgcl 21492 fbasrn 21498 uzfbas 21512 ucnextcn 21918 |
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