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Mirrors > Home > MPE Home > Th. List > fbssint | Structured version Visualization version GIF version |
Description: A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
fbssint | ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fbasne0 23655 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅) | |
2 | n0 4346 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
3 | 1, 2 | sylib 217 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘𝐵) → ∃𝑥 𝑥 ∈ 𝐹) |
4 | ssv 4006 | . . . . . . . 8 ⊢ 𝑥 ⊆ V | |
5 | 4 | jctr 524 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐹 → (𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V)) |
6 | 5 | eximi 1836 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐹 → ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V)) |
7 | df-rex 3070 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ V ↔ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V)) | |
8 | 6, 7 | sylibr 233 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ V) |
9 | 3, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝐵) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ V) |
10 | inteq 4953 | . . . . . . 7 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
11 | int0 4966 | . . . . . . 7 ⊢ ∩ ∅ = V | |
12 | 10, 11 | eqtrdi 2787 | . . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
13 | 12 | sseq2d 4014 | . . . . 5 ⊢ (𝐴 = ∅ → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ⊆ V)) |
14 | 13 | rexbidv 3177 | . . . 4 ⊢ (𝐴 = ∅ → (∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ V)) |
15 | 9, 14 | syl5ibrcom 246 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐴 = ∅ → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)) |
16 | 15 | 3ad2ant1 1132 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)) |
17 | simpl1 1190 | . . . 4 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐹 ∈ (fBas‘𝐵)) | |
18 | simpl2 1191 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐹) | |
19 | simpr 484 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
20 | simpl3 1192 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) | |
21 | elfir 9416 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ (𝐴 ⊆ 𝐹 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐹)) | |
22 | 17, 18, 19, 20, 21 | syl13anc 1371 | . . . 4 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ (fi‘𝐹)) |
23 | fbssfi 23662 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ ∩ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) | |
24 | 17, 22, 23 | syl2anc 583 | . . 3 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) |
25 | 24 | ex 412 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)) |
26 | 16, 25 | pm2.61dne 3027 | 1 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 Vcvv 3473 ⊆ wss 3948 ∅c0 4322 ∩ cint 4950 ‘cfv 6543 Fincfn 8945 ficfi 9411 fBascfbas 21222 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7860 df-1o 8472 df-er 8709 df-en 8946 df-fin 8949 df-fi 9412 df-fbas 21231 |
This theorem is referenced by: fbasfip 23693 |
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