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Mirrors > Home > ILE Home > Th. List > nntri2or2 | Unicode version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.) |
Ref | Expression |
---|---|
nntri2or2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4422 |
. . . . . 6
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2 | 1 | adantl 271 |
. . . . 5
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3 | onelss 4212 |
. . . . 5
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4 | 2, 3 | syl 14 |
. . . 4
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5 | 4 | imp 122 |
. . 3
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6 | 5 | orcd 687 |
. 2
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7 | eqimss 3078 |
. . . 4
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8 | 7 | adantl 271 |
. . 3
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9 | 8 | orcd 687 |
. 2
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10 | nnon 4422 |
. . . . . 6
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11 | 10 | adantr 270 |
. . . . 5
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12 | onelss 4212 |
. . . . 5
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13 | 11, 12 | syl 14 |
. . . 4
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14 | 13 | imp 122 |
. . 3
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15 | 14 | olcd 688 |
. 2
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16 | nntri3or 6246 |
. 2
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17 | 6, 9, 15, 16 | mpjao3dan 1243 |
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 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-iinf 4401 |
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-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 3429 df-sn 3450 df-pr 3451 df-uni 3652 df-int 3687 df-tr 3935 df-iord 4191 df-on 4193 df-suc 4196 df-iom 4404 |
This theorem is referenced by: fientri3 6615 |
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