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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 13572 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀) | |
2 | elfz3 13571 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀)) | |
3 | eleq1 2827 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑀 ∈ (𝑀...𝑀))) | |
4 | 2, 3 | syl5ibrcom 247 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑘 = 𝑀 → 𝑘 ∈ (𝑀...𝑀))) |
5 | 1, 4 | impbid2 226 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 = 𝑀)) |
6 | velsn 4647 | . . 3 ⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) | |
7 | 5, 6 | bitr4di 289 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 ∈ {𝑀})) |
8 | 7 | eqrdv 2733 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 (class class class)co 7431 ℤcz 12611 ...cfz 13544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-fz 13545 |
This theorem is referenced by: fzsuc 13608 fzpred 13609 fzpr 13616 fzsuc2 13619 fz0sn 13664 fz0sn0fz1 13682 fzosn 13772 seqf1o 14081 hashsng 14405 sumsnf 15776 fsum1 15780 fsumm1 15784 fsum1p 15786 prodsn 15995 fprod1 15996 prodsnf 15997 fprod1p 16001 fprodabs 16007 fprodefsum 16128 phi1 16807 vdwlem8 17022 strle1 17192 telgsumfzs 20022 pmatcollpw3fi1 22810 imasdsf1olem 24399 ehl1eudis 25468 voliunlem1 25599 ply1termlem 26257 pntpbnd1 27645 0wlkons1 30150 iuninc 32581 fzspl 32798 esumfzf 34050 ballotlemfc0 34474 ballotlemfcc 34475 plymulx0 34541 signstf0 34562 subfac1 35163 subfacp1lem1 35164 subfacp1lem5 35169 subfacp1lem6 35170 cvmliftlem10 35279 fwddifn0 36146 poimirlem2 37609 poimirlem3 37610 poimirlem4 37611 poimirlem6 37613 poimirlem7 37614 poimirlem13 37620 poimirlem14 37621 poimirlem16 37623 poimirlem17 37624 poimirlem18 37625 poimirlem19 37626 poimirlem20 37627 poimirlem21 37628 poimirlem22 37629 poimirlem26 37633 poimirlem28 37635 poimirlem31 37638 poimirlem32 37639 sdclem1 37730 fdc 37732 aks6d1c1 42098 sticksstones9 42136 sticksstones11 42138 metakunt18 42204 metakunt20 42206 metakunt24 42210 trclfvdecomr 43718 k0004val0 44144 sumsnd 44964 fzdifsuc2 45261 dvnmul 45899 stoweidlem17 45973 carageniuncllem1 46477 caratheodorylem1 46482 hoidmvlelem3 46553 fzopredsuc 47273 sbgoldbo 47712 nnsum3primesprm 47715 stgr1 47864 |
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