![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9141 | . 2 ⊢ 2 < 4 | |
2 | 4lt5 9143 | . 2 ⊢ 4 < 5 | |
3 | 2re 9038 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 4re 9045 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 5re 9047 | . . 3 ⊢ 5 ∈ ℝ | |
6 | 3, 4, 5 | lttri 8110 | . 2 ⊢ ((2 < 4 ∧ 4 < 5) → 2 < 5) |
7 | 1, 2, 6 | mp2an 426 | 1 ⊢ 2 < 5 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4025 < clt 8040 2c2 9019 4c4 9021 5c5 9022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-addcom 7958 ax-addass 7960 ax-i2m1 7963 ax-0lt1 7964 ax-0id 7966 ax-rnegex 7967 ax-pre-lttrn 7972 ax-pre-ltadd 7974 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2758 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-br 4026 df-opab 4087 df-xp 4657 df-iota 5203 df-fv 5250 df-ov 5909 df-pnf 8042 df-mnf 8043 df-ltxr 8045 df-2 9027 df-3 9028 df-4 9029 df-5 9030 |
This theorem is referenced by: scandxnplusgndx 12746 lmodstrd 12755 |
Copyright terms: Public domain | W3C validator |