Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > iscld | Unicode version |
Description: The predicate "the class is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 |
Ref | Expression |
---|---|
iscld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 | |
2 | 1 | cldval 12510 | . . . 4 |
3 | 2 | eleq2d 2227 | . . 3 |
4 | difeq2 3219 | . . . . 5 | |
5 | 4 | eleq1d 2226 | . . . 4 |
6 | 5 | elrab 2868 | . . 3 |
7 | 3, 6 | bitrdi 195 | . 2 |
8 | 1 | topopn 12417 | . . . 4 |
9 | elpw2g 4117 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | 10 | anbi1d 461 | . 2 |
12 | 7, 11 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 crab 2439 cdif 3099 wss 3102 cpw 3543 cuni 3772 cfv 5170 ctop 12406 ccld 12503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-top 12407 df-cld 12506 |
This theorem is referenced by: iscld2 12515 cldss 12516 cldopn 12518 topcld 12520 discld 12547 |
Copyright terms: Public domain | W3C validator |