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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 5166 | . . . . 5 ⊢ 〈0, 3〉 ∈ V | |
2 | opex 5166 | . . . . 5 ⊢ 〈1, 6〉 ∈ V | |
3 | 1, 2 | pm3.2i 464 | . . . 4 ⊢ (〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) |
4 | opex 5166 | . . . . 5 ⊢ 〈0, 2〉 ∈ V | |
5 | opex 5166 | . . . . 5 ⊢ 〈1, 4〉 ∈ V | |
6 | 4, 5 | pm3.2i 464 | . . . 4 ⊢ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V) |
7 | 3, 6 | pm3.2i 464 | . . 3 ⊢ ((〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) ∧ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V)) |
8 | 2re 11453 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 2lt3 11558 | . . . . . . . 8 ⊢ 2 < 3 | |
10 | 8, 9 | gtneii 10490 | . . . . . . 7 ⊢ 3 ≠ 2 |
11 | 10 | olci 855 | . . . . . 6 ⊢ (0 ≠ 0 ∨ 3 ≠ 2) |
12 | c0ex 10372 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 3ex 11462 | . . . . . . 7 ⊢ 3 ∈ V | |
14 | 12, 13 | opthne 5184 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ↔ (0 ≠ 0 ∨ 3 ≠ 2)) |
15 | 11, 14 | mpbir 223 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈0, 2〉 |
16 | 0ne1 11450 | . . . . . . 7 ⊢ 0 ≠ 1 | |
17 | 16 | orci 854 | . . . . . 6 ⊢ (0 ≠ 1 ∨ 3 ≠ 4) |
18 | 12, 13 | opthne 5184 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈1, 4〉 ↔ (0 ≠ 1 ∨ 3 ≠ 4)) |
19 | 17, 18 | mpbir 223 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈1, 4〉 |
20 | 15, 19 | pm3.2i 464 | . . . 4 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) |
21 | 20 | orci 854 | . . 3 ⊢ ((〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) ∨ (〈1, 6〉 ≠ 〈0, 2〉 ∧ 〈1, 6〉 ≠ 〈1, 4〉)) |
22 | prneimg 4618 | . . 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 3035 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ {〈0, 3〉, 〈1, 6〉} ≠ {〈0, 2〉, 〈1, 4〉}) |
27 | 23, 26 | mpbir 223 | 1 ⊢ 𝐴 ≠ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 {cpr 4400 〈cop 4404 (class class class)co 6924 0cc0 10274 1c1 10275 2c2 11434 3c3 11435 4c4 11436 6c6 11438 ℤringzring 20218 freeLMod cfrlm 20493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-2 11442 df-3 11443 |
This theorem is referenced by: zlmodzxzldeplem1 43314 zlmodzxzldeplem3 43316 zlmodzxzldeplem4 43317 ldepsnlinc 43322 |
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