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Mirrors > Home > MPE Home > Th. List > s1cl | Structured version Visualization version GIF 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 | ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 ∈ Word 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 13694 | . 2 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | snopiswrd 13614 | . 2 ⊢ (𝐴 ∈ 𝐵 → {〈0, 𝐴〉} ∈ Word 𝐵) | |
3 | 1, 2 | eqeltrd 2859 | 1 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 ∈ Word 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {csn 4398 〈cop 4404 0cc0 10274 Word cword 13605 〈“cs1 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-word 13606 df-s1 13692 |
This theorem is referenced by: s1cld 13699 s1cli 13701 lsws1 13707 wrdl1s1 13710 ccatws1cl 13712 ccat2s1cl 13714 ccats1alpha 13715 ccat2s1len 13719 ccats1val1 13722 ccats1val2 13723 ccat2s1p1 13725 ccat2s1p2 13726 ccatw2s1ass 13727 lswccats1 13730 cats1un 13847 reuccats1OLD 13854 reuccatpfxs1 13889 s2prop 14064 s2eq2s1eq 14093 s3eqs2s1eq 14095 gsumws2 17776 gsumccatsn 17777 vrmdval 17792 vrmdf 17793 psgnpmtr 18325 wwlksnext 27271 wwlksnextbi 27272 wwlksnextbiOLD 27273 wwlksnextsurj 27281 wwlksnextsurOLD 27286 rusgrnumwwlkb0 27368 loopclwwlkn1b 27449 clwwlkn1loopb 27450 clwwlkext2edg 27470 wwlksext2clwwlk 27471 clwwlknon1loop 27517 1ewlk 27535 1wlkdlem3 27559 numclwwlk2lem1lem 27767 numclwwlk1lem2foalem 27780 numclwwlk1lem2foalemOLD 27781 numclwwlk1lem2fo 27791 numclwwlk1lem2foOLD 27796 signstf0 31253 signstfvn 31254 signstfvp 31257 signstfvneq0 31258 signsvf1 31268 |
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