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Mirrors > Home > MPE Home > Th. List > fz0to3un2pr | Structured version Visualization version GIF version |
Description: An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.) |
Ref | Expression |
---|---|
fz0to3un2pr | ⊢ (0...3) = ({0, 1} ∪ {2, 3}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 11346 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 3nn0 11348 | . . . 4 ⊢ 3 ∈ ℕ0 | |
3 | 1le3 11282 | . . . 4 ⊢ 1 ≤ 3 | |
4 | elfz2nn0 12469 | . . . 4 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
5 | 1, 2, 3, 4 | mpbir3an 1263 | . . 3 ⊢ 1 ∈ (0...3) |
6 | fzsplit 12405 | . . 3 ⊢ (1 ∈ (0...3) → (0...3) = ((0...1) ∪ ((1 + 1)...3))) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (0...3) = ((0...1) ∪ ((1 + 1)...3)) |
8 | 1e0p1 11590 | . . . . 5 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq2i 6701 | . . . 4 ⊢ (0...1) = (0...(0 + 1)) |
10 | 0z 11426 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | fzpr 12434 | . . . . 5 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)}) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ (0...(0 + 1)) = {0, (0 + 1)} |
13 | 0p1e1 11170 | . . . . 5 ⊢ (0 + 1) = 1 | |
14 | 13 | preq2i 4304 | . . . 4 ⊢ {0, (0 + 1)} = {0, 1} |
15 | 9, 12, 14 | 3eqtri 2677 | . . 3 ⊢ (0...1) = {0, 1} |
16 | 1p1e2 11172 | . . . . 5 ⊢ (1 + 1) = 2 | |
17 | df-3 11118 | . . . . 5 ⊢ 3 = (2 + 1) | |
18 | 16, 17 | oveq12i 6702 | . . . 4 ⊢ ((1 + 1)...3) = (2...(2 + 1)) |
19 | 2z 11447 | . . . . 5 ⊢ 2 ∈ ℤ | |
20 | fzpr 12434 | . . . . 5 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)}) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (2...(2 + 1)) = {2, (2 + 1)} |
22 | 2p1e3 11189 | . . . . 5 ⊢ (2 + 1) = 3 | |
23 | 22 | preq2i 4304 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
24 | 18, 21, 23 | 3eqtri 2677 | . . 3 ⊢ ((1 + 1)...3) = {2, 3} |
25 | 15, 24 | uneq12i 3798 | . 2 ⊢ ((0...1) ∪ ((1 + 1)...3)) = ({0, 1} ∪ {2, 3}) |
26 | 7, 25 | eqtri 2673 | 1 ⊢ (0...3) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 ∪ cun 3605 {cpr 4212 class class class wbr 4685 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 ≤ cle 10113 2c2 11108 3c3 11109 ℕ0cn0 11330 ℤcz 11415 ...cfz 12364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 |
This theorem is referenced by: 3wlkdlem4 27140 |
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