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Mirrors > Home > MPE Home > Th. List > 2lt4 | Structured version Visualization version GIF version |
Description: 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
2lt4 | ⊢ 2 < 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt3 12436 | . 2 ⊢ 2 < 3 | |
2 | 3lt4 12438 | . 2 ⊢ 3 < 4 | |
3 | 2re 12338 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 3re 12344 | . . 3 ⊢ 3 ∈ ℝ | |
5 | 4re 12348 | . . 3 ⊢ 4 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11385 | . 2 ⊢ ((2 < 3 ∧ 3 < 4) → 2 < 4) |
7 | 1, 2, 6 | mp2an 692 | 1 ⊢ 2 < 4 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5148 < clt 11293 2c2 12319 3c3 12320 4c4 12321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-2 12327 df-3 12328 df-4 12329 |
This theorem is referenced by: 1lt4 12440 2lt5 12443 eluz4eluz2 12923 fz0to4untppr 13667 fzo0to42pr 13789 4bc2eq6 14365 sqrt2gt1lt2 15310 cos01bnd 16219 4sqlem12 16990 starvndxnplusgndx 17351 prdsvalstr 17499 cnfldfunALTOLDOLD 21411 pcoass 25071 pilem3 26512 ppiublem1 27261 bpos1 27342 2sqlem11 27488 2sqreultlem 27506 2sqreunnltlem 27509 usgrexmplef 29291 upgr4cycl4dv4e 30214 sqsscirc1 33869 iccioo01 37310 hlhilsplusOLD 41926 flt4lem7 42646 fmtno4prmfac 47497 sbgoldbalt 47706 usgrexmpl2lem 47921 usgrexmpl2nb2 47928 usgrexmpl2nb4 47930 usgrexmpl2trifr 47932 |
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