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Mirrors > Home > MPE Home > Th. List > 0ss | Structured version Visualization version GIF version |
Description: The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
0ss | ⊢ ∅ ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3878 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 115 | . 2 ⊢ (𝑥 ∈ ∅ → 𝑥 ∈ 𝐴) |
3 | 2 | ssriv 3572 | 1 ⊢ ∅ ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: ss0b 3925 0pss 3965 npss0 3966 npss0OLD 3967 ssdifeq0 4003 pwpw0 4284 sssn 4298 sspr 4306 sstp 4307 pwsnALT 4367 uni0 4401 int0el 4443 0disj 4575 disjx0 4577 tr0 4692 0elpw 4760 rel0 5166 0ima 5401 dmxpss 5484 dmsnopss 5525 on0eqel 5762 iotassuni 5784 fun0 5868 fresaunres2 5989 f0 5999 fvmptss 6201 fvmptss2 6212 funressn 6331 riotassuni 6547 frxp 7174 suppssdm 7195 suppun 7202 suppss 7212 suppssov1 7214 suppss2 7216 suppssfv 7218 oaword1 7519 oaword2 7520 omwordri 7539 oewordri 7559 oeworde 7560 nnaword1 7596 0domg 7972 fodomr 7996 pwdom 7997 php 8029 isinf 8058 finsschain 8156 fipwuni 8215 fipwss 8218 wdompwdom 8366 inf3lemd 8407 inf3lem1 8408 cantnfle 8451 tc0 8506 r1val1 8532 alephgeom 8788 infmap2 8923 cfub 8954 cf0 8956 cflecard 8958 cfle 8959 fin23lem16 9040 itunitc1 9125 ttukeylem6 9219 ttukeylem7 9220 canthwe 9352 wun0 9419 tsk0 9464 gruina 9519 grur1a 9520 uzssz 11583 xrsup0 12025 fzoss1 12364 fsuppmapnn0fiubex 12654 swrd00 13270 swrdlend 13283 swrd0 13286 repswswrd 13382 xptrrel 13567 sum0 14299 fsumss 14303 fsumcvg3 14307 prod0 14512 0bits 14999 sadid1 15028 sadid2 15029 smu01lem 15045 smu01 15046 smu02 15047 lcmf0 15185 vdwmc2 15521 vdwlem13 15535 ramz2 15566 strfvss 15713 ressbasss 15759 ress0 15761 strlemor0 15795 ismred2 16086 acsfn 16143 acsfn0 16144 0ssc 16320 fullfunc 16389 fthfunc 16390 mrelatglb0 17008 cntzssv 17584 symgsssg 17710 efgsfo 17975 dprdsn 18258 lsp0 18830 lss0v 18837 lspsnat 18966 lsppratlem3 18970 lbsexg 18985 resspsrbas 19236 psr1crng 19378 psr1assa 19379 psr1tos 19380 psr1bas2 19381 vr1cl2 19384 ply1lss 19387 ply1subrg 19388 psr1plusg 19413 psr1vsca 19414 psr1mulr 19415 psr1ring 19438 psr1lmod 19440 psr1sca 19441 evpmss 19751 ocv0 19840 ocvz 19841 css1 19853 0opn 20534 basdif0 20568 baspartn 20569 0cld 20652 cls0 20694 ntr0 20695 cmpfi 21021 refun0 21128 xkouni 21212 xkoccn 21232 alexsubALTlem2 21662 ptcmplem2 21667 tsmsfbas 21741 setsmstopn 22093 restmetu 22185 tngtopn 22264 iccntr 22432 xrge0gsumle 22444 xrge0tsms 22445 metdstri 22462 ovol0 23068 0mbl 23114 itg1le 23286 itgioo 23388 limcnlp 23448 dvbsss 23472 plyssc 23760 fsumharmonic 24538 chocnul 27571 span0 27785 chsup0 27791 ssnnssfz 28937 xrge0tsmsd 29116 ddemeas 29626 dya2iocuni 29672 oms0 29686 0elcarsg 29696 eulerpartlemt 29760 bnj1143 30115 mrsubrn 30664 msubrn 30680 mthmpps 30733 dfpo2 30898 trpredlem1 30971 bj-nuliotaALT 32211 bj-restsn0 32219 bj-restsn10 32220 bj-toponss 32241 mblfinlem2 32617 mblfinlem3 32618 ismblfin 32620 sstotbnd2 32743 isbnd3 32753 ssbnd 32757 heiborlem6 32785 lub0N 33494 glb0N 33498 0psubN 34053 padd01 34115 padd02 34116 pol0N 34213 pcl0N 34226 0psubclN 34247 mzpcompact2lem 36332 itgocn 36753 fvnonrel 36922 clcnvlem 36949 cnvrcl0 36951 cnvtrcl0 36952 0he 37096 ntrclskb 37387 founiiun0 38372 uzfissfz 38483 limcdm0 38685 cncfiooicc 38780 itgvol0 38860 ibliooicc 38863 ovn0 39456 egrsubgr 40501 0grsubgr 40502 0uhgrsubgr 40503 ssnn0ssfz 41920 setrec2fun 42238 |
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