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Mirrors > Home > MPE Home > Th. List > fz0tp | Structured version Visualization version 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 11513 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 1 | addid2i 10626 | . . . 4 ⊢ (0 + 2) = 2 |
3 | 2 | eqcomi 2781 | . . 3 ⊢ 2 = (0 + 2) |
4 | 3 | oveq2i 6985 | . 2 ⊢ (0...2) = (0...(0 + 2)) |
5 | 0z 11802 | . . 3 ⊢ 0 ∈ ℤ | |
6 | fztp 12777 | . . 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 2772 | . . 3 ⊢ 0 = 0 | |
9 | id 22 | . . . 4 ⊢ (0 = 0 → 0 = 0) | |
10 | 0p1e1 11567 | . . . . 5 ⊢ (0 + 1) = 1 | |
11 | 10 | a1i 11 | . . . 4 ⊢ (0 = 0 → (0 + 1) = 1) |
12 | 2 | a1i 11 | . . . 4 ⊢ (0 = 0 → (0 + 2) = 2) |
13 | 9, 11, 12 | tpeq123d 4554 | . . 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 2800 | 1 ⊢ (0...2) = {0, 1, 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 {ctp 4439 (class class class)co 6974 0cc0 10333 1c1 10334 + caddc 10336 2c2 11493 ℤcz 11791 ...cfz 12706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 |
This theorem is referenced by: f13idfv 13181 2wlkdlem4 27446 cshw1s2 30390 |
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