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Mirrors > Home > ILE Home > Th. List > zeo2 | Unicode version |
Description: An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
zeo2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9256 |
. . . . . 6
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2 | peano2cn 8090 |
. . . . . 6
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3 | 1, 2 | syl 14 |
. . . . 5
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4 | 2cnd 8990 |
. . . . 5
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5 | 2ap0 9010 |
. . . . . 6
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6 | 5 | a1i 9 |
. . . . 5
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7 | 3, 4, 6 | divcanap2d 8747 |
. . . 4
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8 | 1, 4, 6 | divcanap2d 8747 |
. . . . 5
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9 | 8 | oveq1d 5889 |
. . . 4
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10 | 7, 9 | eqtr4d 2213 |
. . 3
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11 | zneo 9352 |
. . . . 5
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12 | 11 | expcom 116 |
. . . 4
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13 | 12 | necon2bd 2405 |
. . 3
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14 | 10, 13 | syl5com 29 |
. 2
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15 | zeo 9356 |
. . . 4
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16 | 15 | orcomd 729 |
. . 3
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17 | 16 | ord 724 |
. 2
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18 | 14, 17 | impbid 129 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-n0 9175 df-z 9252 |
This theorem is referenced by: zesq 10635 zeo3 11867 |
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