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Mirrors > Home > MPE Home > Th. List > isfbas | Structured version Visualization version Unicode version |
Description: The predicate " is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
isfbas |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4850 | . . . . 5 | |
2 | elpw2g 4827 | . . . . 5 | |
3 | 1, 2 | syl 17 | . . . 4 |
4 | 3 | anbi1d 741 | . . 3 |
5 | elex 3212 | . . . 4 | |
6 | 5 | biantrurd 529 | . . 3 |
7 | 4, 6 | bitr3d 270 | . 2 |
8 | df-fbas 19743 | . . . 4 | |
9 | neeq1 2856 | . . . . . 6 | |
10 | neleq2 2903 | . . . . . 6 | |
11 | ineq1 3807 | . . . . . . . . 9 | |
12 | 11 | neeq1d 2853 | . . . . . . . 8 |
13 | 12 | raleqbi1dv 3146 | . . . . . . 7 |
14 | 13 | raleqbi1dv 3146 | . . . . . 6 |
15 | 9, 10, 14 | 3anbi123d 1399 | . . . . 5 |
16 | 15 | adantl 482 | . . . 4 |
17 | pweq 4161 | . . . . 5 | |
18 | 17 | pweqd 4163 | . . . 4 |
19 | vpwex 4849 | . . . . . 6 | |
20 | 19 | pwex 4848 | . . . . 5 |
21 | 20 | a1i 11 | . . . 4 |
22 | 8, 16, 18, 21 | elmptrab 21630 | . . 3 |
23 | 3anass 1042 | . . 3 | |
24 | 22, 23 | bitri 264 | . 2 |
25 | 7, 24 | syl6rbbr 279 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wnel 2897 wral 2912 cvv 3200 cin 3573 wss 3574 c0 3915 cpw 4158 cfv 5888 cfbas 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: fbasne0 21634 0nelfb 21635 fbsspw 21636 isfbas2 21639 trfbas2 21647 fbasweak 21669 zfbas 21700 tsmsfbas 21931 ustfilxp 22016 minveclem3b 23199 |
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