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Mirrors > Home > MPE Home > Th. List > s1cl | Structured version Visualization version Unicode version |
Description: A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.) |
Ref | Expression |
---|---|
s1cl | Word |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 13378 | . 2 | |
2 | snopiswrd 13314 | . 2 Word | |
3 | 1, 2 | eqeltrd 2701 | 1 Word |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 csn 4177 cop 4183 cc0 9936 Word cword 13291 cs1 13294 |
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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-nel 2898 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-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-riota 6611 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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-word 13299 df-s1 13302 |
This theorem is referenced by: s1cld 13383 s1cli 13384 lsws1 13391 wrdl1s1 13394 ccatws1cl 13396 ccat2s1cl 13397 ccatws1len 13398 ccat2s1len 13400 ccats1val1 13403 ccats1val2 13404 ccat2s1p1 13405 ccat2s1p2 13406 ccatw2s1ass 13407 lswccats1 13411 cats1un 13475 reuccats1 13480 s2prop 13652 s2eq2s1eq 13681 s3eqs2s1eq 13683 gsumws2 17379 gsumccatsn 17380 vrmdfval 17393 vrmdval 17394 vrmdf 17395 psgnpmtr 17930 wwlksnext 26788 wwlksnextbi 26789 wwlksnextsur 26795 rusgrnumwwlkb0 26866 clwwlksext2edg 26923 wwlksext2clwwlk 26924 1ewlk 26976 1wlkdlem3 26999 clwwlkextfrlem1 27208 numclwlk1lem2foalem 27222 numclwlk1lem2fo 27228 signstf0 30645 signstfvn 30646 signstfvp 30648 signstfvneq0 30649 signsvf1 30658 |
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