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Mirrors > Home > ILE Home > Th. List > 5lt7 | GIF version |
Description: 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
5lt7 | ⊢ 5 < 7 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5lt6 8867 | . 2 ⊢ 5 < 6 | |
2 | 6lt7 8872 | . 2 ⊢ 6 < 7 | |
3 | 5re 8767 | . . 3 ⊢ 5 ∈ ℝ | |
4 | 6re 8769 | . . 3 ⊢ 6 ∈ ℝ | |
5 | 7re 8771 | . . 3 ⊢ 7 ∈ ℝ | |
6 | 3, 4, 5 | lttri 7836 | . 2 ⊢ ((5 < 6 ∧ 6 < 7) → 5 < 7) |
7 | 1, 2, 6 | mp2an 422 | 1 ⊢ 5 < 7 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3899 < clt 7768 5c5 8742 6c6 8743 7c7 8744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 |
This theorem is referenced by: 4lt7 8874 |
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