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Mirrors > Home > ILE Home > Th. List > fz0tp | GIF version |
Description: An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Ref | Expression |
---|---|
fz0tp | ⊢ (0...2) = {0, 1, 2} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 9007 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 1 | addid2i 8117 | . . . 4 ⊢ (0 + 2) = 2 |
3 | 2 | eqcomi 2192 | . . 3 ⊢ 2 = (0 + 2) |
4 | 3 | oveq2i 5901 | . 2 ⊢ (0...2) = (0...(0 + 2)) |
5 | 0z 9281 | . . 3 ⊢ 0 ∈ ℤ | |
6 | fztp 10095 | . . 3 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, (0 + 1), (0 + 2)}) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (0...(0 + 2)) = {0, (0 + 1), (0 + 2)} |
8 | eqid 2188 | . . 3 ⊢ 0 = 0 | |
9 | id 19 | . . . 4 ⊢ (0 = 0 → 0 = 0) | |
10 | 0p1e1 9050 | . . . . 5 ⊢ (0 + 1) = 1 | |
11 | 10 | a1i 9 | . . . 4 ⊢ (0 = 0 → (0 + 1) = 1) |
12 | 2 | a1i 9 | . . . 4 ⊢ (0 = 0 → (0 + 2) = 2) |
13 | 9, 11, 12 | tpeq123d 3698 | . . 3 ⊢ (0 = 0 → {0, (0 + 1), (0 + 2)} = {0, 1, 2}) |
14 | 8, 13 | ax-mp 5 | . 2 ⊢ {0, (0 + 1), (0 + 2)} = {0, 1, 2} |
15 | 4, 7, 14 | 3eqtri 2213 | 1 ⊢ (0...2) = {0, 1, 2} |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2159 {ctp 3608 (class class class)co 5890 0cc0 7828 1c1 7829 + caddc 7831 2c2 8987 ℤcz 9270 ...cfz 10025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-tp 3614 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-inn 8937 df-2 8995 df-n0 9194 df-z 9271 df-uz 9546 df-fz 10026 |
This theorem is referenced by: (None) |
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