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Mirrors > Home > ILE Home > Th. List > 3pos | GIF version |
Description: The number 3 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
3pos | ⊢ 0 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8490 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1re 7485 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 2pos 8511 | . . 3 ⊢ 0 < 2 | |
4 | 0lt1 7608 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 7967 | . 2 ⊢ 0 < (2 + 1) |
6 | df-3 8480 | . 2 ⊢ 3 = (2 + 1) | |
7 | 5, 6 | breqtrri 3870 | 1 ⊢ 0 < 3 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3845 (class class class)co 5652 0cc0 7348 1c1 7349 + caddc 7351 < clt 7520 2c2 8471 3c3 8472 |
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 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-addcom 7443 ax-addass 7445 ax-i2m1 7448 ax-0lt1 7449 ax-0id 7451 ax-rnegex 7452 ax-pre-lttrn 7457 ax-pre-ltadd 7459 |
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-iota 4980 df-fv 5023 df-ov 5655 df-pnf 7522 df-mnf 7523 df-ltxr 7525 df-2 8479 df-3 8480 |
This theorem is referenced by: 3ne0 8515 3ap0 8516 4pos 8517 8th4div3 8633 halfpm6th 8634 sqrt9 10477 ef01bndlem 11043 cos2bnd 11047 sin01gt0 11048 cos01gt0 11049 flodddiv4 11208 ex-gcd 11613 |
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