![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12504 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 3nn0 12506 | . . . 4 ⊢ 3 ∈ ℕ0 | |
3 | 1le3 12440 | . . . 4 ⊢ 1 ≤ 3 | |
4 | elfz2nn0 13610 | . . . 4 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
5 | 1, 2, 3, 4 | mpbir3an 1339 | . . 3 ⊢ 1 ∈ (0...3) |
6 | fzsplit 13545 | . . 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 12735 | . . . . 5 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq2i 7425 | . . . 4 ⊢ (0...1) = (0...(0 + 1)) |
10 | 0z 12585 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | fzpr 13574 | . . . . 5 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)}) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ (0...(0 + 1)) = {0, (0 + 1)} |
13 | 0p1e1 12350 | . . . . 5 ⊢ (0 + 1) = 1 | |
14 | 13 | preq2i 4737 | . . . 4 ⊢ {0, (0 + 1)} = {0, 1} |
15 | 9, 12, 14 | 3eqtri 2759 | . . 3 ⊢ (0...1) = {0, 1} |
16 | 1p1e2 12353 | . . . . 5 ⊢ (1 + 1) = 2 | |
17 | df-3 12292 | . . . . 5 ⊢ 3 = (2 + 1) | |
18 | 16, 17 | oveq12i 7426 | . . . 4 ⊢ ((1 + 1)...3) = (2...(2 + 1)) |
19 | 2z 12610 | . . . . 5 ⊢ 2 ∈ ℤ | |
20 | fzpr 13574 | . . . . 5 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)}) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (2...(2 + 1)) = {2, (2 + 1)} |
22 | 2p1e3 12370 | . . . . 5 ⊢ (2 + 1) = 3 | |
23 | 22 | preq2i 4737 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
24 | 18, 21, 23 | 3eqtri 2759 | . . 3 ⊢ ((1 + 1)...3) = {2, 3} |
25 | 15, 24 | uneq12i 4157 | . 2 ⊢ ((0...1) ∪ ((1 + 1)...3)) = ({0, 1} ∪ {2, 3}) |
26 | 7, 25 | eqtri 2755 | 1 ⊢ (0...3) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∪ cun 3942 {cpr 4626 class class class wbr 5142 (class class class)co 7414 0cc0 11124 1c1 11125 + caddc 11127 ≤ cle 11265 2c2 12283 3c3 12284 ℕ0cn0 12488 ℤcz 12574 ...cfz 13502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 |
This theorem is referenced by: iblcnlem1 25691 3wlkdlem4 29946 |
Copyright terms: Public domain | W3C validator |