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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 14637 | . 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 2106 Word cword 14549 〈“cs1 14630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-word 14550 df-s1 14631 |
This theorem is referenced by: lswccats1fst 14670 ccats1pfxeqbi 14777 cats1cld 14891 cats1co 14892 s2cld 14907 s2co 14956 ofs2 15007 gsumwspan 18872 frmdgsum 18888 frmdss2 18889 frmdup2 18891 gsumwrev 19400 psgnunilem5 19527 efginvrel2 19760 efgs1 19768 efgsp1 19770 efgredlemd 19777 efgredlemc 19778 efgrelexlemb 19783 vrgpf 19801 vrgpinv 19802 frgpup2 19809 frgpup3lem 19810 frgpnabllem1 19906 pgpfaclem1 20116 tgcgr4 28554 clwlkclwwlk2 30032 clwlkclwwlkfo 30038 clwwlkel 30075 clwwlkfo 30079 clwwlkwwlksb 30083 ccatws1f1olast 32922 chnind 32985 chnub 32986 cycpmco2f1 33127 cycpmco2rn 33128 cycpmco2lem2 33130 cycpmco2lem3 33131 cycpmco2lem4 33132 cycpmco2lem5 33133 cycpmco2lem6 33134 cycpmco2lem7 33135 cycpmco2 33136 cyc3genpmlem 33154 elrgspnlem3 33234 unitprodclb 33397 1arithidomlem2 33544 1arithufdlem1 33552 1arithufdlem3 33554 1arithufdlem4 33555 fldext2chn 33734 sseqf 34374 ofcs2 34539 signsvtn 34578 mrsubcv 35495 mrsubff 35497 mrsubrn 35498 mrsubccat 35503 elmrsubrn 35505 mrsubco 35506 mrsubvrs 35507 mvhf 35543 msubvrs 35545 gsumws3 44186 gsumws4 44187 |
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