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Mirrors > Home > MPE Home > Th. List > s1cli | Structured version Visualization version GIF version |
Description: A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1cli | ⊢ 〈“𝐴”〉 ∈ Word V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ids1 13953 | . 2 ⊢ 〈“𝐴”〉 = 〈“( I ‘𝐴)”〉 | |
2 | fvex 6685 | . . 3 ⊢ ( I ‘𝐴) ∈ V | |
3 | s1cl 13958 | . . 3 ⊢ (( I ‘𝐴) ∈ V → 〈“( I ‘𝐴)”〉 ∈ Word V) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈“( I ‘𝐴)”〉 ∈ Word V |
5 | 1, 4 | eqeltri 2911 | 1 ⊢ 〈“𝐴”〉 ∈ Word V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Vcvv 3496 I cid 5461 ‘cfv 6357 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: s1dm 13964 eqs1 13968 ccatws1clv 13973 ccats1alpha 13975 ccatws1len 13976 ccat2s1len 13979 ccats1val1 13983 ccat1st1st 13986 ccat2s1p1 13987 ccat2s1p2 13988 ccatw2s1ass 13991 ccat2s1fvw 14000 revs1 14129 cats1cli 14221 cats1fvn 14222 cats1fv 14223 cats1len 14224 cats1cat 14225 cats2cat 14226 s2cli 14244 s2fv0 14251 s2fv1 14252 s2len 14253 s0s1 14286 s1s2 14287 s1s3 14288 s1s4 14289 s1s5 14290 s1s6 14291 s1s7 14292 s2s2 14293 s4s2 14294 s2s5 14298 s5s2 14299 clwwlkwwlksb 27835 clwwlknon1sn 27881 clwwlknon1le1 27882 1pthon2v 27934 wlk2v2e 27938 konigsberglem1 28033 konigsberglem2 28034 konigsberglem3 28035 loop1cycl 32386 mrsubcv 32759 mrsubrn 32762 mvhf1 32808 msubvrs 32809 |
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