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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 13943 | . 2 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | snopiswrd 13866 | . 2 ⊢ (𝐴 ∈ 𝐵 → {〈0, 𝐴〉} ∈ Word 𝐵) | |
3 | 1, 2 | eqeltrd 2890 | 1 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 ∈ Word 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 {csn 4525 〈cop 4531 0cc0 10526 Word cword 13857 〈“cs1 13940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-word 13858 df-s1 13941 |
This theorem is referenced by: s1cld 13948 s1cli 13950 lsws1 13956 wrdl1s1 13959 ccatws1cl 13961 ccat2s1cl 13963 ccats1alpha 13964 ccat2s1lenOLD 13969 ccats1val1OLD 13973 ccats1val2 13974 ccat2s1p1OLD 13978 ccat2s1p2OLD 13979 ccatw2s1assOLD 13981 lswccats1 13984 cats1un 14074 reuccatpfxs1 14100 s2prop 14260 s2eq2s1eq 14289 s3eqs2s1eq 14291 gsumws2 17999 gsumccatsn 18000 vrmdval 18014 vrmdf 18015 psgnpmtr 18630 efgsval2 18851 wwlksnext 27679 wwlksnextbi 27680 wwlksnextsurj 27686 rusgrnumwwlkb0 27757 loopclwwlkn1b 27827 clwwlkn1loopb 27828 clwwlkext2edg 27841 wwlksext2clwwlk 27842 clwwlknon1loop 27883 1ewlk 27900 1wlkdlem3 27924 numclwwlk2lem1lem 28127 numclwwlk1lem2foalem 28136 numclwwlk1lem2fo 28143 signstf0 31948 signstfvn 31949 signstfvp 31951 signstfvneq0 31952 signstfvc 31954 signsvf1 31961 signsvfn 31962 signshf 31968 |
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