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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 11999 | . 2 ⊢ 2 < 3 | |
2 | 3lt4 12001 | . 2 ⊢ 3 < 4 | |
3 | 2re 11901 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 3re 11907 | . . 3 ⊢ 3 ∈ ℝ | |
5 | 4re 11911 | . . 3 ⊢ 4 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10955 | . 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 5050 < clt 10864 2c2 11882 3c3 11883 4c4 11884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-po 5465 df-so 5466 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-2 11890 df-3 11891 df-4 11892 |
This theorem is referenced by: 1lt4 12003 2lt5 12006 eluz4eluz2 12478 fz0to4untppr 13212 fzo0to42pr 13326 4bc2eq6 13892 sqrt2gt1lt2 14835 cos01bnd 15744 4sqlem12 16506 prdsvalstr 16954 cnfldfun 20372 pcoass 23918 pilem3 25342 ppiublem1 26080 bpos1 26161 2sqlem11 26307 2sqreultlem 26325 2sqreunnltlem 26328 usgrexmplef 27344 upgr4cycl4dv4e 28265 sqsscirc1 31569 iccioo01 35229 hlhilsplus 39689 flt4lem7 40197 fmtno4prmfac 44695 sbgoldbalt 44904 |
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