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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 12223 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀) | |
2 | elfz3 12222 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀)) | |
3 | eleq1 2676 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑀 ∈ (𝑀...𝑀))) | |
4 | 2, 3 | syl5ibrcom 236 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑘 = 𝑀 → 𝑘 ∈ (𝑀...𝑀))) |
5 | 1, 4 | impbid2 215 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 = 𝑀)) |
6 | velsn 4141 | . . 3 ⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) | |
7 | 5, 6 | syl6bbr 277 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 ∈ {𝑀})) |
8 | 7 | eqrdv 2608 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 (class class class)co 6549 ℤcz 11254 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: fzsuc 12258 fzpred 12259 fzpr 12266 fzsuc2 12268 fz0sn 12308 fz0sn0fz1 12325 fzosn 12405 seqf1o 12704 hashsng 13020 sumsn 14319 fsum1 14320 fsumm1 14324 fsum1p 14326 prodsn 14531 fprod1 14532 prodsnf 14533 fprod1p 14537 fprodabs 14543 binomfallfac 14611 ef0lem 14648 fprodefsum 14664 phi1 15316 4sqlem19 15505 vdwlem8 15530 strle1 15800 gsumws1 17199 telgsumfzs 18209 srgbinom 18368 pmatcollpw3fi1lem1 20410 pmatcollpw3fi1 20412 imasdsf1olem 21988 voliunlem1 23125 ply1termlem 23763 pntpbnd1 25075 0spth 26101 eupa0 26501 iuninc 28761 fzspl 28938 esumfzf 29458 ballotlemfc0 29881 ballotlemfcc 29882 plymulx0 29950 signstf0 29971 subfac1 30414 subfacp1lem1 30415 subfacp1lem5 30420 subfacp1lem6 30421 cvmliftlem10 30530 fwddifn0 31441 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem13 32592 poimirlem14 32593 poimirlem16 32595 poimirlem17 32596 poimirlem18 32597 poimirlem19 32598 poimirlem20 32599 poimirlem21 32600 poimirlem22 32601 poimirlem26 32605 poimirlem28 32607 poimirlem31 32610 poimirlem32 32611 sdclem1 32709 fdc 32711 trclfvdecomr 37039 k0004val0 37472 sumsnd 38208 fzdifsuc2 38466 sumsnf 38636 dvnmul 38833 stoweidlem17 38910 carageniuncllem1 39411 caratheodorylem1 39416 hoidmvlelem3 39487 fzopredsuc 39946 nnsum3primesprm 40206 0wlkOns1 41289 0spth-av 41294 |
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