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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 4932 | . . . . 5 ⊢ 〈0, 3〉 ∈ V | |
2 | opex 4932 | . . . . 5 ⊢ 〈1, 6〉 ∈ V | |
3 | 1, 2 | pm3.2i 471 | . . . 4 ⊢ (〈0, 3〉 ∈ V ∧ 〈1, 6〉 ∈ V) |
4 | opex 4932 | . . . . 5 ⊢ 〈0, 2〉 ∈ V | |
5 | opex 4932 | . . . . 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 11090 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 2lt3 11195 | . . . . . . . 8 ⊢ 2 < 3 | |
10 | 8, 9 | gtneii 10149 | . . . . . . 7 ⊢ 3 ≠ 2 |
11 | 10 | olci 406 | . . . . . 6 ⊢ (0 ≠ 0 ∨ 3 ≠ 2) |
12 | c0ex 10034 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 3ex 11096 | . . . . . . 7 ⊢ 3 ∈ V | |
14 | 12, 13 | opthne 4951 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ↔ (0 ≠ 0 ∨ 3 ≠ 2)) |
15 | 11, 14 | mpbir 221 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈0, 2〉 |
16 | 0ne1 11088 | . . . . . . 7 ⊢ 0 ≠ 1 | |
17 | 16 | orci 405 | . . . . . 6 ⊢ (0 ≠ 1 ∨ 3 ≠ 4) |
18 | 12, 13 | opthne 4951 | . . . . . 6 ⊢ (〈0, 3〉 ≠ 〈1, 4〉 ↔ (0 ≠ 1 ∨ 3 ≠ 4)) |
19 | 17, 18 | mpbir 221 | . . . . 5 ⊢ 〈0, 3〉 ≠ 〈1, 4〉 |
20 | 15, 19 | pm3.2i 471 | . . . 4 ⊢ (〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) |
21 | 20 | orci 405 | . . 3 ⊢ ((〈0, 3〉 ≠ 〈0, 2〉 ∧ 〈0, 3〉 ≠ 〈1, 4〉) ∨ (〈1, 6〉 ≠ 〈0, 2〉 ∧ 〈1, 6〉 ≠ 〈1, 4〉)) |
22 | prneimg 4388 | . . 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 2860 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ {〈0, 3〉, 〈1, 6〉} ≠ {〈0, 2〉, 〈1, 4〉}) |
27 | 23, 26 | mpbir 221 | 1 ⊢ 𝐴 ≠ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 {cpr 4179 〈cop 4183 (class class class)co 6650 0cc0 9936 1c1 9937 2c2 11070 3c3 11071 4c4 11072 6c6 11074 ℤringzring 19818 freeLMod cfrlm 20090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-2 11079 df-3 11080 |
This theorem is referenced by: zlmodzxzldeplem1 42289 zlmodzxzldeplem3 42291 zlmodzxzldeplem4 42292 ldepsnlinc 42297 |
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