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Mirrors > Home > MPE Home > Th. List > fzo0to42pr | Structured version Visualization version GIF version |
Description: A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
Ref | Expression |
---|---|
fzo0to42pr | ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11993 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 11995 | . . . 4 ⊢ 4 ∈ ℕ0 | |
3 | 2re 11790 | . . . . 5 ⊢ 2 ∈ ℝ | |
4 | 4re 11800 | . . . . 5 ⊢ 4 ∈ ℝ | |
5 | 2lt4 11891 | . . . . 5 ⊢ 2 < 4 | |
6 | 3, 4, 5 | ltleii 10841 | . . . 4 ⊢ 2 ≤ 4 |
7 | elfz2nn0 13089 | . . . 4 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ0 ∧ 4 ∈ ℕ0 ∧ 2 ≤ 4)) | |
8 | 1, 2, 6, 7 | mpbir3an 1342 | . . 3 ⊢ 2 ∈ (0...4) |
9 | fzosplit 13161 | . . 3 ⊢ (2 ∈ (0...4) → (0..^4) = ((0..^2) ∪ (2..^4))) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ (0..^4) = ((0..^2) ∪ (2..^4)) |
11 | fzo0to2pr 13213 | . . 3 ⊢ (0..^2) = {0, 1} | |
12 | 4z 12097 | . . . . 5 ⊢ 4 ∈ ℤ | |
13 | fzoval 13130 | . . . . 5 ⊢ (4 ∈ ℤ → (2..^4) = (2...(4 − 1))) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (2..^4) = (2...(4 − 1)) |
15 | 4m1e3 11845 | . . . . . . 7 ⊢ (4 − 1) = 3 | |
16 | df-3 11780 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
17 | 15, 16 | eqtri 2761 | . . . . . 6 ⊢ (4 − 1) = (2 + 1) |
18 | 17 | oveq2i 7181 | . . . . 5 ⊢ (2...(4 − 1)) = (2...(2 + 1)) |
19 | 2z 12095 | . . . . . 6 ⊢ 2 ∈ ℤ | |
20 | fzpr 13053 | . . . . . 6 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (2...(2 + 1)) = {2, (2 + 1)} |
22 | 18, 21 | eqtri 2761 | . . . 4 ⊢ (2...(4 − 1)) = {2, (2 + 1)} |
23 | 2p1e3 11858 | . . . . 5 ⊢ (2 + 1) = 3 | |
24 | 23 | preq2i 4628 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
25 | 14, 22, 24 | 3eqtri 2765 | . . 3 ⊢ (2..^4) = {2, 3} |
26 | 11, 25 | uneq12i 4051 | . 2 ⊢ ((0..^2) ∪ (2..^4)) = ({0, 1} ∪ {2, 3}) |
27 | 10, 26 | eqtri 2761 | 1 ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 ∪ cun 3841 {cpr 4518 class class class wbr 5030 (class class class)co 7170 0cc0 10615 1c1 10616 + caddc 10618 ≤ cle 10754 − cmin 10948 2c2 11771 3c3 11772 4c4 11773 ℕ0cn0 11976 ℤcz 12062 ...cfz 12981 ..^cfzo 13124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-fzo 13125 |
This theorem is referenced by: 3pthdlem1 28101 upgr4cycl4dv4e 28122 |
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