Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8783 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1 | ltp1i 8656 | . 2 ⊢ 2 < (2 + 1) |
3 | df-3 8773 | . 2 ⊢ 3 = (2 + 1) | |
4 | 2, 3 | breqtrri 3950 | 1 ⊢ 2 < 3 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3924 (class class class)co 5767 1c1 7614 + caddc 7616 < clt 7793 2c2 8764 3c3 8765 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-2 8772 df-3 8773 |
This theorem is referenced by: 1lt3 8884 2lt4 8886 2lt6 8895 2lt7 8901 2lt8 8908 2lt9 8916 3halfnz 9141 2lt10 9312 uzuzle23 9359 uz3m2nn 9361 fztpval 9856 expnass 10391 cos01gt0 11458 3lcm2e6 11827 plusgndxnmulrndx 12061 rngstrg 12063 coseq00topi 12905 coseq0negpitopi 12906 cos02pilt1 12921 ex-fl 12926 |
Copyright terms: Public domain | W3C validator |