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Mirrors > Home > ILE Home > Th. List > 2lt5 | GIF version |
Description: 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
2lt5 | ⊢ 2 < 5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt4 8745 | . 2 ⊢ 2 < 4 | |
2 | 4lt5 8747 | . 2 ⊢ 4 < 5 | |
3 | 2re 8648 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 4re 8655 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 5re 8657 | . . 3 ⊢ 5 ∈ ℝ | |
6 | 3, 4, 5 | lttri 7739 | . 2 ⊢ ((2 < 4 ∧ 4 < 5) → 2 < 5) |
7 | 1, 2, 6 | mp2an 420 | 1 ⊢ 2 < 5 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3875 < clt 7672 2c2 8629 4c4 8631 5c5 8632 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-i2m1 7600 ax-0lt1 7601 ax-0id 7603 ax-rnegex 7604 ax-pre-lttrn 7609 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-iota 5024 df-fv 5067 df-ov 5709 df-pnf 7674 df-mnf 7675 df-ltxr 7677 df-2 8637 df-3 8638 df-4 8639 df-5 8640 |
This theorem is referenced by: lmodstrd 11874 |
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