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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 6220 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ∈ dom fBas) | |
2 | isfbas 21633 | . . . . 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 256 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))) |
5 | simpr2 1068 | . . 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 2898 | . 2 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹) | |
8 | 6, 7 | sylib 208 | 1 ⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∀wral 2912 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 dom cdm 5114 ‘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: fbdmn0 21638 fbncp 21643 fbun 21644 fbfinnfr 21645 0nelfil 21653 fsubbas 21671 fbasfip 21672 fgcl 21682 fbasrn 21688 uzfbas 21702 ucnextcn 22108 |
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