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Mirrors > Home > MPE Home > Th. List > 4lt9 | Structured version Visualization version GIF version |
Description: 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
4lt9 | ⊢ 4 < 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4lt5 12004 | . 2 ⊢ 4 < 5 | |
2 | 5lt9 12029 | . 2 ⊢ 5 < 9 | |
3 | 4re 11911 | . . 3 ⊢ 4 ∈ ℝ | |
4 | 5re 11914 | . . 3 ⊢ 5 ∈ ℝ | |
5 | 9re 11926 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10955 | . 2 ⊢ ((4 < 5 ∧ 5 < 9) → 4 < 9) |
7 | 1, 2, 6 | mp2an 692 | 1 ⊢ 4 < 9 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5050 < clt 10864 4c4 11884 5c5 11885 9c9 11889 |
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 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 |
This theorem is referenced by: 3lt9 12031 cnfldstr 20362 cnfldfun 20372 341fppr2 44857 9fppr8 44860 bgoldbnnsum3prm 44927 |
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