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Mirrors > Home > MPE Home > Th. List > fileln0 | Structured version Visualization version GIF version |
Description: An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
fileln0 | ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ 𝐹) | |
2 | 0nelfil 23773 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹) | |
3 | nelne2 3037 | . 2 ⊢ ((𝐴 ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐹) → 𝐴 ≠ ∅) | |
4 | 1, 2, 3 | syl2anr 595 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2937 ∅c0 4326 ‘cfv 6553 Filcfil 23769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fv 6561 df-fbas 21283 df-fil 23770 |
This theorem is referenced by: filinn0 23784 filintn0 23785 alexsublem 23968 cfil3i 25217 iscmet3 25241 filnetlem4 35898 |
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