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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 14574 | . 2 ⊢ (𝐴 ∈ 𝐵 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | snopiswrd 14499 | . 2 ⊢ (𝐴 ∈ 𝐵 → {⟨0, 𝐴⟩} ∈ Word 𝐵) | |
3 | 1, 2 | eqeltrd 2829 | 1 ⊢ (𝐴 ∈ 𝐵 → ⟨“𝐴”⟩ ∈ Word 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 {csn 4624 ⟨cop 4630 0cc0 11132 Word cword 14490 ⟨“cs1 14571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-word 14491 df-s1 14572 |
This theorem is referenced by: s1cld 14579 s1cli 14581 lsws1 14587 wrdl1s1 14590 ccatws1cl 14592 ccat2s1cl 14594 ccats1alpha 14595 ccats1val2 14603 lswccats1 14610 cats1un 14697 reuccatpfxs1 14723 s2prop 14884 s2eq2s1eq 14913 s3eqs2s1eq 14915 gsumws2 18787 gsumccatsn 18788 vrmdval 18802 vrmdf 18803 psgnpmtr 19458 efgsval2 19681 wwlksnext 29697 wwlksnextbi 29698 wwlksnextsurj 29704 rusgrnumwwlkb0 29775 loopclwwlkn1b 29845 clwwlkn1loopb 29846 clwwlkext2edg 29859 wwlksext2clwwlk 29860 clwwlknon1loop 29901 1ewlk 29918 1wlkdlem3 29942 numclwwlk2lem1lem 30145 numclwwlk1lem2foalem 30154 numclwwlk1lem2fo 30161 signstf0 34194 signstfvn 34195 signstfvp 34197 signstfvneq0 34198 signstfvc 34200 signsvf1 34207 signsvfn 34208 signshf 34214 upwordsing 46264 |
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