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Mirrors > Home > MPE Home > Th. List > connsubclo | Structured version Visualization version Unicode version |
Description: If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
connsubclo.1 | |
connsubclo.3 | |
connsubclo.4 | ↾t Conn |
connsubclo.5 | |
connsubclo.6 | |
connsubclo.7 |
Ref | Expression |
---|---|
connsubclo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ↾t ↾t | |
2 | connsubclo.4 | . . . 4 ↾t Conn | |
3 | connsubclo.7 | . . . . . 6 | |
4 | cldrcl 20830 | . . . . . 6 | |
5 | 3, 4 | syl 17 | . . . . 5 |
6 | connsubclo.1 | . . . . . . . 8 | |
7 | 6 | topopn 20711 | . . . . . . 7 |
8 | 5, 7 | syl 17 | . . . . . 6 |
9 | connsubclo.3 | . . . . . 6 | |
10 | 8, 9 | ssexd 4805 | . . . . 5 |
11 | connsubclo.5 | . . . . 5 | |
12 | elrestr 16089 | . . . . 5 ↾t | |
13 | 5, 10, 11, 12 | syl3anc 1326 | . . . 4 ↾t |
14 | connsubclo.6 | . . . 4 | |
15 | eqid 2622 | . . . . . 6 | |
16 | ineq1 3807 | . . . . . . . 8 | |
17 | 16 | eqeq2d 2632 | . . . . . . 7 |
18 | 17 | rspcev 3309 | . . . . . 6 |
19 | 3, 15, 18 | sylancl 694 | . . . . 5 |
20 | 6 | restcld 20976 | . . . . . 6 ↾t |
21 | 5, 9, 20 | syl2anc 693 | . . . . 5 ↾t |
22 | 19, 21 | mpbird 247 | . . . 4 ↾t |
23 | 1, 2, 13, 14, 22 | connclo 21218 | . . 3 ↾t |
24 | 6 | restuni 20966 | . . . 4 ↾t |
25 | 5, 9, 24 | syl2anc 693 | . . 3 ↾t |
26 | 23, 25 | eqtr4d 2659 | . 2 |
27 | sseqin2 3817 | . 2 | |
28 | 26, 27 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 wne 2794 wrex 2913 cvv 3200 cin 3573 wss 3574 c0 3915 cuni 4436 cfv 5888 (class class class)co 6650 ↾t crest 16081 ctop 20698 ccld 20820 Conncconn 21214 |
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-rep 4771 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-3or 1038 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-reu 2919 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-conn 21215 |
This theorem is referenced by: conncn 21229 conncompclo 21238 |
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