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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 13958 | . 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 2114 Word cword 13864 〈“cs1 13951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-word 13865 df-s1 13952 |
This theorem is referenced by: lswccats1fst 13996 ccats1pfxeqbi 14106 cats1cld 14219 cats1co 14220 s2cld 14235 s2co 14284 ofs2 14333 gsumwspan 18013 frmdgsum 18029 frmdss2 18030 frmdup2 18032 gsumwrev 18496 psgnunilem5 18624 efginvrel2 18855 efgs1 18863 efgsp1 18865 efgredlemd 18872 efgredlemc 18873 efgrelexlemb 18878 vrgpf 18896 vrgpinv 18897 frgpup2 18904 frgpup3lem 18905 frgpnabllem1 18995 pgpfaclem1 19205 tgcgr4 26319 wlklenvclwlkOLD 27439 clwlkclwwlk2 27783 clwlkclwwlkfo 27789 clwwlkel 27827 clwwlkfo 27831 clwwlkwwlksb 27835 cycpmco2f1 30768 cycpmco2rn 30769 cycpmco2lem2 30771 cycpmco2lem3 30772 cycpmco2lem4 30773 cycpmco2lem5 30774 cycpmco2lem6 30775 cycpmco2lem7 30776 cycpmco2 30777 cyc3genpmlem 30795 sseqf 31652 ofcs2 31817 signsvtn 31856 mrsubcv 32759 mrsubff 32761 mrsubrn 32762 mrsubccat 32767 elmrsubrn 32769 mrsubco 32770 mrsubvrs 32771 mvhf 32807 msubvrs 32809 gsumws3 40556 gsumws4 40557 |
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