![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2lt10 | Structured version Visualization version GIF version |
Description: 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
2lt10 | ⊢ 2 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt3 12371 | . 2 ⊢ 2 < 3 | |
2 | 3lt10 12801 | . 2 ⊢ 3 < ;10 | |
3 | 2re 12273 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 3re 12279 | . . 3 ⊢ 3 ∈ ℝ | |
5 | 10re 12683 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11327 | . 2 ⊢ ((2 < 3 ∧ 3 < ;10) → 2 < ;10) |
7 | 1, 2, 6 | mp2an 691 | 1 ⊢ 2 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5144 0cc0 11097 1c1 11098 < clt 11235 2c2 12254 3c3 12255 ;cdc 12664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-dec 12665 |
This theorem is referenced by: 1lt10 12803 163prm 17045 631prm 17047 4001prm 17065 plendxnplusgndx 17303 dsndxnplusgndx 17322 slotsdifunifndx 17333 mgpdsOLD 19984 cnfldfunALTOLD 20932 znaddOLD 21068 opsrplusgOLD 21577 slotsinbpsd 27659 slotslnbpsd 27660 ttgplusgOLD 28100 oppgleOLD 32102 hgt750lem 33594 lcmineqlem 40823 3lexlogpow5ineq1 40825 3lexlogpow5ineq5 40831 aks4d1p1 40847 257prm 46102 127prm 46140 |
Copyright terms: Public domain | W3C validator |