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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 13508 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀) | |
2 | elfz3 13507 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀)) | |
3 | eleq1 2821 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑀 ∈ (𝑀...𝑀))) | |
4 | 2, 3 | syl5ibrcom 246 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑘 = 𝑀 → 𝑘 ∈ (𝑀...𝑀))) |
5 | 1, 4 | impbid2 225 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 = 𝑀)) |
6 | velsn 4643 | . . 3 ⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) | |
7 | 5, 6 | bitr4di 288 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 ∈ {𝑀})) |
8 | 7 | eqrdv 2730 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4627 (class class class)co 7405 ℤcz 12554 ...cfz 13480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-neg 11443 df-z 12555 df-uz 12819 df-fz 13481 |
This theorem is referenced by: fzsuc 13544 fzpred 13545 fzpr 13552 fzsuc2 13555 fz0sn 13597 fz0sn0fz1 13614 fzosn 13699 seqf1o 14005 hashsng 14325 sumsnf 15685 fsum1 15689 fsumm1 15693 fsum1p 15695 prodsn 15902 fprod1 15903 prodsnf 15904 fprod1p 15908 fprodabs 15914 fprodefsum 16034 phi1 16702 vdwlem8 16917 strle1 17087 telgsumfzs 19851 pmatcollpw3fi1 22281 imasdsf1olem 23870 ehl1eudis 24928 voliunlem1 25058 ply1termlem 25708 pntpbnd1 27078 0wlkons1 29363 iuninc 31779 fzspl 31988 esumfzf 33055 ballotlemfc0 33479 ballotlemfcc 33480 plymulx0 33546 signstf0 33567 subfac1 34157 subfacp1lem1 34158 subfacp1lem5 34163 subfacp1lem6 34164 cvmliftlem10 34273 fwddifn0 35124 poimirlem2 36478 poimirlem3 36479 poimirlem4 36480 poimirlem6 36482 poimirlem7 36483 poimirlem13 36489 poimirlem14 36490 poimirlem16 36492 poimirlem17 36493 poimirlem18 36494 poimirlem19 36495 poimirlem20 36496 poimirlem21 36497 poimirlem22 36498 poimirlem26 36502 poimirlem28 36504 poimirlem31 36507 poimirlem32 36508 sdclem1 36599 fdc 36601 sticksstones9 40958 sticksstones11 40960 metakunt18 40990 metakunt20 40992 metakunt24 40996 trclfvdecomr 42464 k0004val0 42890 sumsnd 43695 fzdifsuc2 44006 dvnmul 44645 stoweidlem17 44719 carageniuncllem1 45223 caratheodorylem1 45228 hoidmvlelem3 45299 fzopredsuc 46017 sbgoldbo 46441 nnsum3primesprm 46444 |
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