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Mirrors > Home > ILE Home > Th. List > isnzr2 | Unicode version |
Description: Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
isnzr2.b |
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Ref | Expression |
---|---|
isnzr2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2193 |
. . 3
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2 | eqid 2193 |
. . 3
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3 | 1, 2 | isnzr 13661 |
. 2
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4 | isnzr2.b |
. . . . . . . . 9
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5 | 4, 1 | ringidcl 13500 |
. . . . . . . 8
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6 | 5 | adantr 276 |
. . . . . . 7
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7 | 4, 2 | ring0cl 13501 |
. . . . . . . 8
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8 | 7 | adantr 276 |
. . . . . . 7
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9 | simpr 110 |
. . . . . . 7
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10 | df-ne 2365 |
. . . . . . . . 9
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11 | neeq1 2377 |
. . . . . . . . 9
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12 | 10, 11 | bitr3id 194 |
. . . . . . . 8
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13 | neeq2 2378 |
. . . . . . . 8
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14 | 12, 13 | rspc2ev 2879 |
. . . . . . 7
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15 | 6, 8, 9, 14 | syl3anc 1249 |
. . . . . 6
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16 | 15 | ex 115 |
. . . . 5
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17 | 4, 1, 2 | ring1eq0 13528 |
. . . . . . . 8
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18 | 17 | 3expb 1206 |
. . . . . . 7
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19 | 18 | necon3bd 2407 |
. . . . . 6
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20 | 19 | rexlimdvva 2619 |
. . . . 5
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21 | 16, 20 | impbid 129 |
. . . 4
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22 | simpl 109 |
. . . . . . . . . . . 12
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23 | simprl 529 |
. . . . . . . . . . . 12
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24 | simprr 531 |
. . . . . . . . . . . 12
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25 | 22, 23, 24 | enpr2d 6862 |
. . . . . . . . . . 11
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26 | 25 | adantl 277 |
. . . . . . . . . 10
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27 | 26 | ensymd 6828 |
. . . . . . . . 9
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28 | basfn 12666 |
. . . . . . . . . . . . 13
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29 | elex 2771 |
. . . . . . . . . . . . 13
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30 | funfvex 5563 |
. . . . . . . . . . . . . 14
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31 | 30 | funfni 5346 |
. . . . . . . . . . . . 13
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32 | 28, 29, 31 | sylancr 414 |
. . . . . . . . . . . 12
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33 | 4, 32 | eqeltrid 2280 |
. . . . . . . . . . 11
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34 | ssdomg 6823 |
. . . . . . . . . . 11
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35 | 33, 34 | syl 14 |
. . . . . . . . . 10
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36 | 22, 23 | prssd 3777 |
. . . . . . . . . 10
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37 | 35, 36 | impel 280 |
. . . . . . . . 9
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38 | endomtr 6835 |
. . . . . . . . 9
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39 | 27, 37, 38 | syl2anc 411 |
. . . . . . . 8
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40 | 39 | anassrs 400 |
. . . . . . 7
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41 | 40 | rexlimdvaa 2612 |
. . . . . 6
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42 | 41 | rexlimdva 2611 |
. . . . 5
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43 | 2dom 6850 |
. . . . 5
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44 | 42, 43 | impbid1 142 |
. . . 4
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45 | 21, 44 | bitrd 188 |
. . 3
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46 | 45 | pm5.32i 454 |
. 2
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47 | 3, 46 | bitri 184 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-addcom 7962 ax-addass 7964 ax-i2m1 7967 ax-0lt1 7968 ax-0id 7970 ax-rnegex 7971 ax-pre-ltirr 7974 ax-pre-ltadd 7978 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-iord 4395 df-on 4397 df-suc 4400 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1o 6460 df-2o 6461 df-er 6578 df-en 6786 df-dom 6787 df-pnf 8046 df-mnf 8047 df-ltxr 8049 df-inn 8973 df-2 9031 df-3 9032 df-ndx 12611 df-slot 12612 df-base 12614 df-sets 12615 df-plusg 12698 df-mulr 12699 df-0g 12859 df-mgm 12929 df-sgrp 12975 df-mnd 12988 df-grp 13065 df-minusg 13066 df-mgp 13401 df-ur 13440 df-ring 13478 df-nzr 13660 |
This theorem is referenced by: znidom 14122 znidomb 14123 |
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