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Mirrors > Home > MPE Home > Th. List > eltg3 | Structured version Visualization version Unicode version |
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
eltg3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6220 | . . . 4 | |
2 | inex1g 4801 | . . . 4 | |
3 | 1, 2 | syl 17 | . . 3 |
4 | eltg4i 20764 | . . 3 | |
5 | inss1 3833 | . . . . . . 7 | |
6 | sseq1 3626 | . . . . . . 7 | |
7 | 5, 6 | mpbiri 248 | . . . . . 6 |
8 | 7 | biantrurd 529 | . . . . 5 |
9 | unieq 4444 | . . . . . 6 | |
10 | 9 | eqeq2d 2632 | . . . . 5 |
11 | 8, 10 | bitr3d 270 | . . . 4 |
12 | 11 | spcegv 3294 | . . 3 |
13 | 3, 4, 12 | sylc 65 | . 2 |
14 | eltg3i 20765 | . . . . 5 | |
15 | eleq1 2689 | . . . . 5 | |
16 | 14, 15 | syl5ibrcom 237 | . . . 4 |
17 | 16 | expimpd 629 | . . 3 |
18 | 17 | exlimdv 1861 | . 2 |
19 | 13, 18 | impbid2 216 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 cin 3573 wss 3574 cpw 4158 cuni 4436 cdm 5114 cfv 5888 ctg 16098 |
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 ax-un 6949 |
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-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-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 |
This theorem is referenced by: tgval3 20767 tgtop 20777 eltop3 20780 tgidm 20784 bastop1 20797 tgrest 20963 tgcn 21056 txbasval 21409 opnmblALT 23371 mbfimaopnlem 23422 isfne3 32338 fneuni 32342 dissneqlem 33187 tgqioo2 39774 |
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