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Mirrors > Home > MPE Home > Th. List > 2lt5 | Structured version Visualization version GIF version |
Description: 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
2lt5 | ⊢ 2 < 5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt4 12018 | . 2 ⊢ 2 < 4 | |
2 | 4lt5 12020 | . 2 ⊢ 4 < 5 | |
3 | 2re 11917 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 4re 11927 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 5re 11930 | . . 3 ⊢ 5 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10971 | . 2 ⊢ ((2 < 4 ∧ 4 < 5) → 2 < 5) |
7 | 1, 2, 6 | mp2an 692 | 1 ⊢ 2 < 5 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5062 < clt 10880 2c2 11898 4c4 11900 5c5 11901 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-po 5477 df-so 5478 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-er 8400 df-en 8636 df-dom 8637 df-sdom 8638 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-2 11906 df-3 11907 df-4 11908 df-5 11909 |
This theorem is referenced by: 37prm 16687 317prm 16692 lmodstr 16874 mgpsca 19524 sraaddg 20229 zlmplusg 20498 resvplusg 31264 12gcd5e1 39758 257prm 44701 127prm 44739 |
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