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Mirrors > Home > ILE Home > Th. List > 2lt3 | GIF version |
Description: 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
Ref | Expression |
---|---|
2lt3 | ⊢ 2 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8935 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1 | ltp1i 8808 | . 2 ⊢ 2 < (2 + 1) |
3 | df-3 8925 | . 2 ⊢ 3 = (2 + 1) | |
4 | 2, 3 | breqtrri 4014 | 1 ⊢ 2 < 3 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3987 (class class class)co 5850 1c1 7762 + caddc 7764 < clt 7941 2c2 8916 3c3 8917 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-iota 5158 df-fv 5204 df-ov 5853 df-pnf 7943 df-mnf 7944 df-ltxr 7946 df-2 8924 df-3 8925 |
This theorem is referenced by: 1lt3 9036 2lt4 9038 2lt6 9047 2lt7 9053 2lt8 9060 2lt9 9068 3halfnz 9296 2lt10 9467 uzuzle23 9517 uz3m2nn 9519 fztpval 10026 expnass 10568 cos01gt0 11712 3lcm2e6 12101 plusgndxnmulrndx 12517 rngstrg 12519 coseq00topi 13471 coseq0negpitopi 13472 cos02pilt1 13487 2logb9irr 13604 2logb3irr 13606 2logb9irrap 13610 ex-fl 13681 |
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