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Mirrors > Home > MPE Home > Th. List > s1cld | Structured version Visualization version GIF version |
Description: A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1cld.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
s1cld | ⊢ (𝜑 → 〈“𝐴”〉 ∈ Word 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | s1cl 13592 | . 2 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 ∈ Word 𝐵) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 〈“𝐴”〉 ∈ Word 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 Word cword 13497 〈“cs1 13500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-fzo 12680 df-word 13505 df-s1 13508 |
This theorem is referenced by: eqs1 13603 lswccats1fst 13631 ccats1swrdeqbi 13718 cats1cld 13820 cats1co 13821 s2cld 13836 s2co 13885 ofs2 13931 gsumwspan 17604 frmdgsum 17620 frmdss2 17621 frmdup2 17623 gsumwrev 18016 psgnunilem5 18134 efginvrel2 18360 efgsval2 18366 efgs1 18368 efgsp1 18370 efgredlemd 18377 efgredlemc 18378 efgrelexlemb 18383 vrgpf 18401 vrgpinv 18402 frgpup2 18409 frgpup3lem 18410 frgpnabllem1 18496 pgpfaclem1 18700 tgcgr4 25646 wlklenvclwlk 26782 clwlkclwwlk2 27147 clwlkclwwlkfo 27153 clwwlkel 27196 clwwlkfo 27200 clwwlkwwlksb 27205 clwlksfoclwwlkOLD 27238 sseqf 30784 ofcs2 30952 signstfvneq0 30979 signstfvc 30981 signsvfn 30989 signsvtn 30991 signshf 30995 mrsubcv 31735 mrsubff 31737 mrsubrn 31738 mrsubccat 31743 elmrsubrn 31745 mrsubco 31746 mrsubvrs 31747 mvhf 31783 msubvrs 31785 gsumws3 39019 gsumws4 39020 ccats1pfxeqbi 41959 |
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