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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 13755 | . 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 2048 Word cword 13662 〈“cs1 13748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-fzo 12843 df-word 13663 df-s1 13749 |
This theorem is referenced by: eqs1 13765 lswccats1fst 13788 ccats1pfxeqbi 13939 ccats1swrdeqbiOLD 13940 cats1cld 14069 cats1co 14070 s2cld 14085 s2co 14134 ofs2 14182 gsumwspan 17842 frmdgsum 17858 frmdss2 17859 frmdup2 17861 gsumwrev 18255 psgnunilem5 18373 psgnunilem5OLD 18374 efginvrel2 18601 efgsval2 18607 efgs1 18609 efgsp1 18611 efgredlemd 18619 efgredlemc 18620 efgrelexlemb 18626 vrgpf 18644 vrgpinv 18645 frgpup2 18652 frgpup3lem 18653 frgpnabllem1 18739 pgpfaclem1 18943 tgcgr4 26009 wlklenvclwlk 27129 clwlkclwwlk2 27500 clwlkclwwlk2OLD 27501 clwlkclwwlkfoOLD 27509 clwlkclwwlkfo 27513 clwwlkel 27553 clwwlkfoOLD 27557 clwwlkfo 27562 clwwlkwwlksb 27567 sseqf 31253 ofcs2 31422 signstfvneq0 31450 signstfvc 31452 signsvfn 31461 signsvtn 31463 signshf 31467 mrsubcv 32217 mrsubff 32219 mrsubrn 32220 mrsubccat 32225 elmrsubrn 32227 mrsubco 32228 mrsubvrs 32229 mvhf 32265 msubvrs 32267 gsumws3 39859 gsumws4 39860 |
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