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Mirrors > Home > ILE Home > Th. List > p1le | Unicode version |
Description: A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
Ref | Expression |
---|---|
p1le |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lep1 8306 |
. . . 4
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2 | 1 | adantr 270 |
. . 3
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3 | peano2re 7618 |
. . . . 5
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4 | 3 | ancli 316 |
. . . 4
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5 | letr 7568 |
. . . . 5
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6 | 5 | 3expa 1143 |
. . . 4
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7 | 4, 6 | sylan 277 |
. . 3
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8 | 2, 7 | mpand 420 |
. 2
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9 | 8 | 3impia 1140 |
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-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-addcom 7445 ax-addass 7447 ax-i2m1 7450 ax-0lt1 7451 ax-0id 7453 ax-rnegex 7454 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-ltadd 7461 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-cnv 4446 df-iota 4980 df-fv 5023 df-ov 5655 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 |
This theorem is referenced by: fzind 8861 |
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