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Mirrors > Home > MPE Home > Th. List > ufilss | Structured version Visualization version Unicode version |
Description: For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.) |
Ref | Expression |
---|---|
ufilss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6220 | . . . 4 | |
2 | elpw2g 4827 | . . . 4 | |
3 | 1, 2 | syl 17 | . . 3 |
4 | isufil 21707 | . . . . 5 | |
5 | 4 | simprbi 480 | . . . 4 |
6 | eleq1 2689 | . . . . . 6 | |
7 | difeq2 3722 | . . . . . . 7 | |
8 | 7 | eleq1d 2686 | . . . . . 6 |
9 | 6, 8 | orbi12d 746 | . . . . 5 |
10 | 9 | rspccv 3306 | . . . 4 |
11 | 5, 10 | syl 17 | . . 3 |
12 | 3, 11 | sylbird 250 | . 2 |
13 | 12 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 wral 2912 cdif 3571 wss 3574 cpw 4158 cdm 5114 cfv 5888 cfil 21649 cufil 21703 |
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-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-ufil 21705 |
This theorem is referenced by: ufilb 21710 trufil 21714 ufildr 21735 |
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