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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 13451 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀) | |
2 | elfz3 13450 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀)) | |
3 | eleq1 2825 | . . . . 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 4602 | . . 3 ⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) | |
7 | 5, 6 | bitr4di 288 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 ∈ {𝑀})) |
8 | 7 | eqrdv 2734 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4586 (class class class)co 7356 ℤcz 12498 ...cfz 13423 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-pre-lttri 11124 ax-pre-lttrn 11125 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-neg 11387 df-z 12499 df-uz 12763 df-fz 13424 |
This theorem is referenced by: fzsuc 13487 fzpred 13488 fzpr 13495 fzsuc2 13498 fz0sn 13540 fz0sn0fz1 13557 fzosn 13642 seqf1o 13948 hashsng 14268 sumsnf 15627 fsum1 15631 fsumm1 15635 fsum1p 15637 prodsn 15844 fprod1 15845 prodsnf 15846 fprod1p 15850 fprodabs 15856 fprodefsum 15976 phi1 16644 vdwlem8 16859 strle1 17029 telgsumfzs 19764 pmatcollpw3fi1 22135 imasdsf1olem 23724 ehl1eudis 24782 voliunlem1 24912 ply1termlem 25562 pntpbnd1 26932 0wlkons1 29012 iuninc 31426 fzspl 31637 esumfzf 32608 ballotlemfc0 33032 ballotlemfcc 33033 plymulx0 33099 signstf0 33120 subfac1 33712 subfacp1lem1 33713 subfacp1lem5 33718 subfacp1lem6 33719 cvmliftlem10 33828 fwddifn0 34739 poimirlem2 36070 poimirlem3 36071 poimirlem4 36072 poimirlem6 36074 poimirlem7 36075 poimirlem13 36081 poimirlem14 36082 poimirlem16 36084 poimirlem17 36085 poimirlem18 36086 poimirlem19 36087 poimirlem20 36088 poimirlem21 36089 poimirlem22 36090 poimirlem26 36094 poimirlem28 36096 poimirlem31 36099 poimirlem32 36100 sdclem1 36192 fdc 36194 sticksstones9 40552 sticksstones11 40554 metakunt18 40584 metakunt20 40586 metakunt24 40590 trclfvdecomr 41981 k0004val0 42407 sumsnd 43212 fzdifsuc2 43519 dvnmul 44155 stoweidlem17 44229 carageniuncllem1 44733 caratheodorylem1 44738 hoidmvlelem3 44809 fzopredsuc 45526 sbgoldbo 45950 nnsum3primesprm 45953 |
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