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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 13952 | . 2 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | snopiswrd 13871 | . 2 ⊢ (𝐴 ∈ 𝐵 → {〈0, 𝐴〉} ∈ Word 𝐵) | |
3 | 1, 2 | eqeltrd 2913 | 1 ⊢ (𝐴 ∈ 𝐵 → 〈“𝐴”〉 ∈ Word 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {csn 4567 〈cop 4573 0cc0 10537 Word cword 13862 〈“cs1 13949 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-word 13863 df-s1 13950 |
This theorem is referenced by: s1cld 13957 s1cli 13959 lsws1 13965 wrdl1s1 13968 ccatws1cl 13970 ccat2s1cl 13972 ccats1alpha 13973 ccat2s1lenOLD 13978 ccats1val1OLD 13982 ccats1val2 13983 ccat2s1p1OLD 13987 ccat2s1p2OLD 13988 ccatw2s1assOLD 13990 lswccats1 13993 cats1un 14083 reuccatpfxs1 14109 s2prop 14269 s2eq2s1eq 14298 s3eqs2s1eq 14300 gsumws2 18007 gsumccatsn 18008 vrmdval 18022 vrmdf 18023 psgnpmtr 18638 efgsval2 18859 wwlksnext 27671 wwlksnextbi 27672 wwlksnextsurj 27678 rusgrnumwwlkb0 27750 loopclwwlkn1b 27820 clwwlkn1loopb 27821 clwwlkext2edg 27835 wwlksext2clwwlk 27836 clwwlknon1loop 27877 1ewlk 27894 1wlkdlem3 27918 numclwwlk2lem1lem 28121 numclwwlk1lem2foalem 28130 numclwwlk1lem2fo 28137 signstf0 31838 signstfvn 31839 signstfvp 31841 signstfvneq0 31842 signstfvc 31844 signsvf1 31851 signsvfn 31852 signshf 31858 |
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