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Mirrors > Home > MPE Home > Th. List > fzsn | Structured version Visualization version GIF version |
Description: A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
fzsn | ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz1eq 12919 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀) | |
2 | elfz3 12918 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀)) | |
3 | eleq1 2900 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑀 ∈ (𝑀...𝑀))) | |
4 | 2, 3 | syl5ibrcom 249 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑘 = 𝑀 → 𝑘 ∈ (𝑀...𝑀))) |
5 | 1, 4 | impbid2 228 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 = 𝑀)) |
6 | velsn 4583 | . . 3 ⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) | |
7 | 5, 6 | syl6bbr 291 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 ∈ {𝑀})) |
8 | 7 | eqrdv 2819 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4567 (class class class)co 7156 ℤcz 11982 ...cfz 12893 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
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-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 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-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 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-neg 10873 df-z 11983 df-uz 12245 df-fz 12894 |
This theorem is referenced by: fzsuc 12955 fzpred 12956 fzpr 12963 fzsuc2 12966 fz0sn 13008 fz0sn0fz1 13025 fzosn 13109 seqf1o 13412 hashsng 13731 sumsnf 15099 fsum1 15102 fsumm1 15106 fsum1p 15108 prodsn 15316 fprod1 15317 prodsnf 15318 fprod1p 15322 fprodabs 15328 fprodefsum 15448 phi1 16110 vdwlem8 16324 strle1 16592 telgsumfzs 19109 pmatcollpw3fi1 21396 imasdsf1olem 22983 ehl1eudis 24023 voliunlem1 24151 ply1termlem 24793 pntpbnd1 26162 0wlkons1 27900 iuninc 30312 fzspl 30513 esumfzf 31328 ballotlemfc0 31750 ballotlemfcc 31751 plymulx0 31817 signstf0 31838 subfac1 32425 subfacp1lem1 32426 subfacp1lem5 32431 subfacp1lem6 32432 cvmliftlem10 32541 fwddifn0 33625 poimirlem2 34909 poimirlem3 34910 poimirlem4 34911 poimirlem6 34913 poimirlem7 34914 poimirlem13 34920 poimirlem14 34921 poimirlem16 34923 poimirlem17 34924 poimirlem18 34925 poimirlem19 34926 poimirlem20 34927 poimirlem21 34928 poimirlem22 34929 poimirlem26 34933 poimirlem28 34935 poimirlem31 34938 poimirlem32 34939 sdclem1 35033 fdc 35035 trclfvdecomr 40122 k0004val0 40553 sumsnd 41332 fzdifsuc2 41626 dvnmul 42277 stoweidlem17 42351 carageniuncllem1 42852 caratheodorylem1 42857 hoidmvlelem3 42928 fzopredsuc 43572 sbgoldbo 44001 nnsum3primesprm 44004 |
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