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Mirrors > Home > ILE Home > Th. List > sscls | Unicode version |
Description: A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.) |
Ref | Expression |
---|---|
clscld.1 |
Ref | Expression |
---|---|
sscls |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 3759 | . 2 | |
2 | clscld.1 | . . 3 | |
3 | 2 | clsval 12207 | . 2 |
4 | 1, 3 | sseqtrrid 3118 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 crab 2397 wss 3041 cuni 3706 cint 3741 cfv 5093 ctop 12091 ccld 12188 ccl 12190 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-top 12092 df-cld 12191 df-cls 12193 |
This theorem is referenced by: ntrcls0 12227 |
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