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| Mirrors > Home > MPE Home > Th. List > fbncp | Structured version Visualization version GIF version | ||
| Description: A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
| Ref | Expression |
|---|---|
| fbncp | ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0nelfb 23839 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ ∅ ∈ 𝐹) |
| 3 | fbasssin 23844 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴))) | |
| 4 | disjdif 4472 | . . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
| 5 | 4 | sseq2i 4013 | . . . . . . 7 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) ↔ 𝑥 ⊆ ∅) |
| 6 | ss0 4402 | . . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
| 7 | 5, 6 | sylbi 217 | . . . . . 6 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → 𝑥 = ∅) |
| 8 | eleq1 2829 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐹 ↔ ∅ ∈ 𝐹)) | |
| 9 | 8 | biimpac 478 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 = ∅) → ∅ ∈ 𝐹) |
| 10 | 7, 9 | sylan2 593 | . . . . 5 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴))) → ∅ ∈ 𝐹) |
| 11 | 10 | rexlimiva 3147 | . . . 4 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → ∅ ∈ 𝐹) |
| 12 | 3, 11 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∅ ∈ 𝐹) |
| 13 | 12 | 3expia 1122 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐵 ∖ 𝐴) ∈ 𝐹 → ∅ ∈ 𝐹)) |
| 14 | 2, 13 | mtod 198 | 1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 fBascfbas 21352 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-fbas 21361 |
| This theorem is referenced by: filconn 23891 fgtr 23898 ufilb 23914 ufilmax 23915 ufilen 23938 flimrest 23991 fclsrest 24032 cfilres 25330 relcmpcmet 25352 |
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