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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 5469 | . . . . 5 ⊢ 〈0, 3〉 ∈ V | |
| 2 | opex 5469 | . . . . 5 ⊢ 〈1, 6〉 ∈ V | |
| 3 | 1, 2 | pm3.2i 470 | . . . 4 ⊢ (〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) |
| 4 | opex 5469 | . . . . 5 ⊢ 〈0, 2〉 ∈ V | |
| 5 | opex 5469 | . . . . 5 ⊢ 〈1, 4〉 ∈ V | |
| 6 | 4, 5 | pm3.2i 470 | . . . 4 ⊢ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V) |
| 7 | 3, 6 | pm3.2i 470 | . . 3 ⊢ ((〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) ∧ (〈0, 2〉 ∈ V ∧ 〈1, 4〉 ∈ V)) |
| 8 | 2re 12340 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
| 9 | 2lt3 12438 | . . . . . . . 8 ⊢ 2 < 3 | |
| 10 | 8, 9 | gtneii 11373 | . . . . . . 7 ⊢ 3 ≠ 2 |
| 11 | 10 | olci 867 | . . . . . 6 ⊢ (0 ≠ 0 ∨ 3 ≠ 2) |
| 12 | c0ex 11255 | . . . . . . 7 ⊢ 0 ∈ V | |
| 13 | 3ex 12348 | . . . . . . 7 ⊢ 3 ∈ V | |
| 14 | 12, 13 | opthne 5487 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ↔ (0 ≠ 0 ∨ 3 ≠ 2)) |
| 15 | 11, 14 | mpbir 231 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈0, 2〉 |
| 16 | 0ne1 12337 | . . . . . . 7 ⊢ 0 ≠ 1 | |
| 17 | 16 | orci 866 | . . . . . 6 ⊢ (0 ≠ 1 ∨ 3 ≠ 4) |
| 18 | 12, 13 | opthne 5487 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈1, 4〉 ↔ (0 ≠ 1 ∨ 3 ≠ 4)) |
| 19 | 17, 18 | mpbir 231 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈1, 4〉 |
| 20 | 15, 19 | pm3.2i 470 | . . . 4 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) |
| 21 | 20 | orci 866 | . . 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 231 | 1 ⊢ 𝐴 ≠ 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {cpr 4628 〈cop 4632 (class class class)co 7431 0cc0 11155 1c1 11156 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℤringczring 21457 freeLMod cfrlm 21766 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-2 12329 df-3 12330 |
| This theorem is referenced by: zlmodzxzldeplem1 48417 zlmodzxzldeplem3 48419 zlmodzxzldeplem4 48420 ldepsnlinc 48425 |
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