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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem | Structured version Visualization version GIF version |
Description: A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxzldep.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxzldep.a | ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} |
zlmodzxzldep.b | ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} |
Ref | Expression |
---|---|
zlmodzxzldeplem | ⊢ 𝐴 ≠ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5463 | . . . . 5 ⊢ 〈0, 3〉 ∈ V | |
2 | opex 5463 | . . . . 5 ⊢ 〈1, 6〉 ∈ V | |
3 | 1, 2 | pm3.2i 471 | . . . 4 ⊢ (〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) |
4 | opex 5463 | . . . . 5 ⊢ 〈0, 2〉 ∈ V | |
5 | opex 5463 | . . . . 5 ⊢ 〈1, 4〉 ∈ V | |
6 | 4, 5 | pm3.2i 471 | . . . 4 ⊢ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V) |
7 | 3, 6 | pm3.2i 471 | . . 3 ⊢ ((〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) ∧ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V)) |
8 | 2re 12282 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 2lt3 12380 | . . . . . . . 8 ⊢ 2 < 3 | |
10 | 8, 9 | gtneii 11322 | . . . . . . 7 ⊢ 3 ≠ 2 |
11 | 10 | olci 864 | . . . . . 6 ⊢ (0 ≠ 0 ∨ 3 ≠ 2) |
12 | c0ex 11204 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 3ex 12290 | . . . . . . 7 ⊢ 3 ∈ V | |
14 | 12, 13 | opthne 5481 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ↔ (0 ≠ 0 ∨ 3 ≠ 2)) |
15 | 11, 14 | mpbir 230 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈0, 2〉 |
16 | 0ne1 12279 | . . . . . . 7 ⊢ 0 ≠ 1 | |
17 | 16 | orci 863 | . . . . . 6 ⊢ (0 ≠ 1 ∨ 3 ≠ 4) |
18 | 12, 13 | opthne 5481 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈1, 4〉 ↔ (0 ≠ 1 ∨ 3 ≠ 4)) |
19 | 17, 18 | mpbir 230 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈1, 4〉 |
20 | 15, 19 | pm3.2i 471 | . . . 4 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) |
21 | 20 | orci 863 | . . 3 ⊢ ((〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) ∨ (〈1, 6〉 ≠ 〈0, 2〉 ∧ 〈1, 6〉 ≠ 〈1, 4〉)) |
22 | prneimg 4854 | . . 3 ⊢ (((〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) ∧ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V)) → (((〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) ∨ (〈1, 6〉 ≠ 〈0, 2〉 ∧ 〈1, 6〉 ≠ 〈1, 4〉)) → {〈0, 3〉, 〈1, 6〉} ≠ {〈0, 2〉, 〈1, 4〉})) | |
23 | 7, 21, 22 | mp2 9 | . 2 ⊢ {〈0, 3〉, 〈1, 6〉} ≠ {〈0, 2〉, 〈1, 4〉} |
24 | zlmodzxzldep.a | . . 3 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
25 | zlmodzxzldep.b | . . 3 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
26 | 24, 25 | neeq12i 3007 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ {〈0, 3〉, 〈1, 6〉} ≠ {〈0, 2〉, 〈1, 4〉}) |
27 | 23, 26 | mpbir 230 | 1 ⊢ 𝐴 ≠ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 {cpr 4629 〈cop 4633 (class class class)co 7405 0cc0 11106 1c1 11107 2c2 12263 3c3 12264 4c4 12265 6c6 12267 ℤringczring 21009 freeLMod cfrlm 21292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-2 12271 df-3 12272 |
This theorem is referenced by: zlmodzxzldeplem1 47134 zlmodzxzldeplem3 47136 zlmodzxzldeplem4 47137 ldepsnlinc 47142 |
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