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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 13453 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀) | |
2 | elfz3 13452 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (𝑀...𝑀)) | |
3 | eleq1 2826 | . . . . 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 225 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 = 𝑀)) |
6 | velsn 4603 | . . 3 ⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) | |
7 | 5, 6 | bitr4di 289 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ (𝑀...𝑀) ↔ 𝑘 ∈ {𝑀})) |
8 | 7 | eqrdv 2735 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4587 (class class class)co 7358 ℤcz 12500 ...cfz 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-neg 11389 df-z 12501 df-uz 12765 df-fz 13426 |
This theorem is referenced by: fzsuc 13489 fzpred 13490 fzpr 13497 fzsuc2 13500 fz0sn 13542 fz0sn0fz1 13559 fzosn 13644 seqf1o 13950 hashsng 14270 sumsnf 15629 fsum1 15633 fsumm1 15637 fsum1p 15639 prodsn 15846 fprod1 15847 prodsnf 15848 fprod1p 15852 fprodabs 15858 fprodefsum 15978 phi1 16646 vdwlem8 16861 strle1 17031 telgsumfzs 19767 pmatcollpw3fi1 22140 imasdsf1olem 23729 ehl1eudis 24787 voliunlem1 24917 ply1termlem 25567 pntpbnd1 26937 0wlkons1 29068 iuninc 31482 fzspl 31696 esumfzf 32671 ballotlemfc0 33095 ballotlemfcc 33096 plymulx0 33162 signstf0 33183 subfac1 33775 subfacp1lem1 33776 subfacp1lem5 33781 subfacp1lem6 33782 cvmliftlem10 33891 fwddifn0 34752 poimirlem2 36083 poimirlem3 36084 poimirlem4 36085 poimirlem6 36087 poimirlem7 36088 poimirlem13 36094 poimirlem14 36095 poimirlem16 36097 poimirlem17 36098 poimirlem18 36099 poimirlem19 36100 poimirlem20 36101 poimirlem21 36102 poimirlem22 36103 poimirlem26 36107 poimirlem28 36109 poimirlem31 36112 poimirlem32 36113 sdclem1 36205 fdc 36207 sticksstones9 40565 sticksstones11 40567 metakunt18 40597 metakunt20 40599 metakunt24 40603 trclfvdecomr 42007 k0004val0 42433 sumsnd 43238 fzdifsuc2 43551 dvnmul 44191 stoweidlem17 44265 carageniuncllem1 44769 caratheodorylem1 44774 hoidmvlelem3 44845 fzopredsuc 45562 sbgoldbo 45986 nnsum3primesprm 45989 |
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