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Mirrors > Home > ILE Home > Th. List > nntri1 | Unicode version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) |
Ref | Expression |
---|---|
nntri1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnel 4385 |
. 2
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2 | nntri3or 6254 |
. . . 4
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3 | df-3or 925 |
. . . . . . 7
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4 | 3 | biimpi 118 |
. . . . . 6
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5 | 4 | orcomd 683 |
. . . . 5
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6 | 5 | ord 678 |
. . . 4
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7 | 2, 6 | syl 14 |
. . 3
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8 | nnord 4426 |
. . . . . . 7
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9 | ordelss 4206 |
. . . . . . 7
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10 | 8, 9 | sylan 277 |
. . . . . 6
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11 | 10 | ex 113 |
. . . . 5
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12 | 11 | adantl 271 |
. . . 4
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13 | eqimss 3078 |
. . . . 5
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14 | 13 | a1i 9 |
. . . 4
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15 | 12, 14 | jaod 672 |
. . 3
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16 | 7, 15 | syld 44 |
. 2
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17 | 1, 16 | impbid2 141 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-tr 3937 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 |
This theorem is referenced by: nnsseleq 6262 nnmword 6275 nnawordex 6285 nndomo 6578 nninfalllemn 11853 |
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