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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 9019 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1 | ltp1i 8892 | . 2 ⊢ 2 < (2 + 1) |
3 | df-3 9009 | . 2 ⊢ 3 = (2 + 1) | |
4 | 2, 3 | breqtrri 4045 | 1 ⊢ 2 < 3 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4018 (class class class)co 5896 1c1 7842 + caddc 7844 < clt 8022 2c2 9000 3c3 9001 |
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 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-pre-ltadd 7957 |
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 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-iota 5196 df-fv 5243 df-ov 5899 df-pnf 8024 df-mnf 8025 df-ltxr 8027 df-2 9008 df-3 9009 |
This theorem is referenced by: 1lt3 9120 2lt4 9122 2lt6 9131 2lt7 9137 2lt8 9144 2lt9 9152 3halfnz 9380 2lt10 9551 uzuzle23 9601 uz3m2nn 9603 fztpval 10113 expnass 10657 cos01gt0 11802 3lcm2e6 12192 plusgndxnmulrndx 12644 rngstrg 12646 slotsdifunifndx 12739 coseq00topi 14713 coseq0negpitopi 14714 cos02pilt1 14729 2logb9irr 14846 2logb3irr 14848 2logb9irrap 14852 ex-fl 14935 |
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