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Mirrors > Home > ILE Home > Th. List > fzo0to2pr | GIF version |
Description: A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Ref | Expression |
---|---|
fzo0to2pr | ⊢ (0..^2) = {0, 1} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9075 | . . 3 ⊢ 2 ∈ ℤ | |
2 | fzoval 9918 | . . 3 ⊢ (2 ∈ ℤ → (0..^2) = (0...(2 − 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (0..^2) = (0...(2 − 1)) |
4 | 2m1e1 8831 | . . . 4 ⊢ (2 − 1) = 1 | |
5 | 0p1e1 8827 | . . . 4 ⊢ (0 + 1) = 1 | |
6 | 4, 5 | eqtr4i 2161 | . . 3 ⊢ (2 − 1) = (0 + 1) |
7 | 6 | oveq2i 5778 | . 2 ⊢ (0...(2 − 1)) = (0...(0 + 1)) |
8 | 0z 9058 | . . 3 ⊢ 0 ∈ ℤ | |
9 | fzpr 9850 | . . . 4 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)}) | |
10 | 5 | preq2i 3599 | . . . 4 ⊢ {0, (0 + 1)} = {0, 1} |
11 | 9, 10 | syl6eq 2186 | . . 3 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, 1}) |
12 | 8, 11 | ax-mp 5 | . 2 ⊢ (0...(0 + 1)) = {0, 1} |
13 | 3, 7, 12 | 3eqtri 2162 | 1 ⊢ (0..^2) = {0, 1} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cpr 3523 (class class class)co 5767 0cc0 7613 1c1 7614 + caddc 7616 − cmin 7926 2c2 8764 ℤcz 9047 ...cfz 9783 ..^cfzo 9912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-fzo 9913 |
This theorem is referenced by: fzo0to42pr 9990 |
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