![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s1cli | Structured version Visualization version GIF version |
Description: A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1cli | ⊢ 〈“𝐴”〉 ∈ Word V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ids1 13617 | . 2 ⊢ 〈“𝐴”〉 = 〈“( I ‘𝐴)”〉 | |
2 | fvex 6424 | . . 3 ⊢ ( I ‘𝐴) ∈ V | |
3 | s1cl 13622 | . . 3 ⊢ (( I ‘𝐴) ∈ V → 〈“( I ‘𝐴)”〉 ∈ Word V) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈“( I ‘𝐴)”〉 ∈ Word V |
5 | 1, 4 | eqeltri 2874 | 1 ⊢ 〈“𝐴”〉 ∈ Word V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 Vcvv 3385 I cid 5219 ‘cfv 6101 Word cword 13534 〈“cs1 13615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-word 13535 df-s1 13616 |
This theorem is referenced by: s1dm 13628 ccatws1clv 13637 ccats1alpha 13639 ccatws1len 13640 ccat1st1st 13651 revs1 13845 cats1cli 13942 cats1fvn 13943 cats1fv 13944 cats1len 13945 cats1cat 13946 cats2cat 13947 s2cli 13965 s2fv0 13972 s2fv1 13973 s2len 13974 s0s1 14007 s1s2 14008 s1s3 14009 s1s4 14010 s1s5 14011 s1s6 14012 s1s7 14013 s2s2 14014 s4s2 14015 s2s5 14019 s5s2 14020 clwwlkwwlksb 27370 clwwlknon1sn 27439 clwwlknon1le1 27440 1pthon2v 27497 wlk2v2e 27501 konigsberglem1 27599 konigsberglem2 27600 konigsberglem3 27601 mrsubcv 31924 mrsubrn 31927 mvhf1 31973 msubvrs 31974 |
Copyright terms: Public domain | W3C validator |