Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0nelfil | Structured version Visualization version GIF version |
Description: The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
0nelfil | ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filfbas 22458 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | 0nelfb 22441 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 ∅c0 4293 ‘cfv 6357 fBascfbas 20535 Filcfil 22455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-fbas 20544 df-fil 22456 |
This theorem is referenced by: fileln0 22460 isfil2 22466 infil 22473 filuni 22495 filufint 22530 rnelfmlem 22562 fmfnfm 22568 fclscmpi 22639 |
Copyright terms: Public domain | W3C validator |