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Mirrors > Home > ILE Home > Th. List > oddnn02np1 | Unicode version |
Description: A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
oddnn02np1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2145 |
. . . . . . . 8
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2 | elnn0z 8659 |
. . . . . . . . 9
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3 | 2tnp1ge0ge0 9597 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | biimpd 142 |
. . . . . . . . . . . 12
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5 | 4 | imdistani 434 |
. . . . . . . . . . 11
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6 | 5 | expcom 114 |
. . . . . . . . . 10
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7 | elnn0z 8659 |
. . . . . . . . . 10
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8 | 6, 7 | syl6ibr 160 |
. . . . . . . . 9
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9 | 2, 8 | simplbiim 379 |
. . . . . . . 8
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10 | 1, 9 | syl6bir 162 |
. . . . . . 7
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11 | 10 | com13 79 |
. . . . . 6
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12 | 11 | impcom 123 |
. . . . 5
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13 | 12 | pm4.71rd 386 |
. . . 4
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14 | 13 | bicomd 139 |
. . 3
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15 | 14 | rexbidva 2371 |
. 2
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16 | nn0ssz 8664 |
. . 3
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17 | rexss 3072 |
. . 3
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18 | 16, 17 | mp1i 10 |
. 2
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19 | nn0z 8666 |
. . 3
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20 | odd2np1 10653 |
. . 3
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21 | 19, 20 | syl 14 |
. 2
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22 | 15, 18, 21 | 3bitr4rd 219 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-xor 1308 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-po 4087 df-iso 4088 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-2 8375 df-n0 8566 df-z 8647 df-dvds 10577 |
This theorem is referenced by: oddge22np1 10661 |
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