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Mirrors > Home > MPE Home > Th. List > 1lt7 | Structured version Visualization version GIF version |
Description: 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
1lt7 | ⊢ 1 < 7 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2 12172 | . 2 ⊢ 1 < 2 | |
2 | 2lt7 12191 | . 2 ⊢ 2 < 7 | |
3 | 1re 11003 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 12075 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 7re 12094 | . . 3 ⊢ 7 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11129 | . 2 ⊢ ((1 < 2 ∧ 2 < 7) → 1 < 7) |
7 | 1, 2, 6 | mp2an 688 | 1 ⊢ 1 < 7 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5077 1c1 10900 < clt 11037 2c2 12056 7c7 12061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 |
This theorem is referenced by: 7prm 16840 prmlem2 16849 43prm 16851 317prm 16855 631prm 16856 aks4d1p1p2 40104 127prm 45091 |
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