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Mirrors > Home > ILE Home > Th. List > nn0lt2 | Unicode version |
Description: A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
Ref | Expression |
---|---|
nn0lt2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 665 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | nn0z 8680 |
. . . . . 6
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4 | 2z 8688 |
. . . . . 6
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5 | zltlem1 8717 |
. . . . . 6
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6 | 3, 4, 5 | sylancl 404 |
. . . . 5
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7 | 2m1e1 8451 |
. . . . . 6
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8 | 7 | breq2i 3822 |
. . . . 5
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9 | 6, 8 | syl6bb 194 |
. . . 4
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10 | necom 2335 |
. . . . 5
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11 | 1z 8686 |
. . . . . . . 8
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12 | zltlen 8735 |
. . . . . . . 8
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13 | 3, 11, 12 | sylancl 404 |
. . . . . . 7
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14 | nn0lt10b 8737 |
. . . . . . . . . 10
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15 | 14 | biimpa 290 |
. . . . . . . . 9
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16 | 15 | orcd 685 |
. . . . . . . 8
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17 | 16 | ex 113 |
. . . . . . 7
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18 | 13, 17 | sylbird 168 |
. . . . . 6
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19 | 18 | expd 254 |
. . . . 5
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20 | 10, 19 | syl7bi 163 |
. . . 4
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21 | 9, 20 | sylbid 148 |
. . 3
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22 | 21 | imp 122 |
. 2
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23 | zdceq 8732 |
. . . . 5
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24 | 3, 11, 23 | sylancl 404 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | adantr 270 |
. . 3
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26 | dcne 2262 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | sylib 120 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 2, 22, 27 | mpjaod 671 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-mulrcl 7365 ax-addcom 7366 ax-mulcom 7367 ax-addass 7368 ax-mulass 7369 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-1rid 7373 ax-0id 7374 ax-rnegex 7375 ax-precex 7376 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-apti 7381 ax-pre-ltadd 7382 ax-pre-mulgt0 7383 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-br 3815 df-opab 3869 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-reap 7970 df-ap 7977 df-inn 8335 df-2 8393 df-n0 8584 df-z 8661 |
This theorem is referenced by: (None) |
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