![]() |
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 12430 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 3nn0 12432 | . . . 4 ⊢ 3 ∈ ℕ0 | |
3 | 1le3 12366 | . . . 4 ⊢ 1 ≤ 3 | |
4 | elfz2nn0 13533 | . . . 4 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
5 | 1, 2, 3, 4 | mpbir3an 1342 | . . 3 ⊢ 1 ∈ (0...3) |
6 | fzsplit 13468 | . . 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 12661 | . . . . 5 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq2i 7369 | . . . 4 ⊢ (0...1) = (0...(0 + 1)) |
10 | 0z 12511 | . . . . 5 ⊢ 0 ∈ ℤ | |
11 | fzpr 13497 | . . . . 5 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)}) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ (0...(0 + 1)) = {0, (0 + 1)} |
13 | 0p1e1 12276 | . . . . 5 ⊢ (0 + 1) = 1 | |
14 | 13 | preq2i 4699 | . . . 4 ⊢ {0, (0 + 1)} = {0, 1} |
15 | 9, 12, 14 | 3eqtri 2769 | . . 3 ⊢ (0...1) = {0, 1} |
16 | 1p1e2 12279 | . . . . 5 ⊢ (1 + 1) = 2 | |
17 | df-3 12218 | . . . . 5 ⊢ 3 = (2 + 1) | |
18 | 16, 17 | oveq12i 7370 | . . . 4 ⊢ ((1 + 1)...3) = (2...(2 + 1)) |
19 | 2z 12536 | . . . . 5 ⊢ 2 ∈ ℤ | |
20 | fzpr 13497 | . . . . 5 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)}) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (2...(2 + 1)) = {2, (2 + 1)} |
22 | 2p1e3 12296 | . . . . 5 ⊢ (2 + 1) = 3 | |
23 | 22 | preq2i 4699 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
24 | 18, 21, 23 | 3eqtri 2769 | . . 3 ⊢ ((1 + 1)...3) = {2, 3} |
25 | 15, 24 | uneq12i 4122 | . 2 ⊢ ((0...1) ∪ ((1 + 1)...3)) = ({0, 1} ∪ {2, 3}) |
26 | 7, 25 | eqtri 2765 | 1 ⊢ (0...3) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {cpr 4589 class class class wbr 5106 (class class class)co 7358 0cc0 11052 1c1 11053 + caddc 11055 ≤ cle 11191 2c2 12209 3c3 12210 ℕ0cn0 12414 ℤcz 12500 ...cfz 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 |
This theorem is referenced by: iblcnlem1 25155 3wlkdlem4 29109 |
Copyright terms: Public domain | W3C validator |