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Mirrors > Home > MPE Home > Th. List > 2lt9 | Structured version Visualization version GIF version |
Description: 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
2lt9 | ⊢ 2 < 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt3 11803 | . 2 ⊢ 2 < 3 | |
2 | 3lt9 11835 | . 2 ⊢ 3 < 9 | |
3 | 2re 11705 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 3re 11711 | . . 3 ⊢ 3 ∈ ℝ | |
5 | 9re 11730 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10760 | . 2 ⊢ ((2 < 3 ∧ 3 < 9) → 2 < 9) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ 2 < 9 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5059 < clt 10669 2c2 11686 3c3 11687 9c9 11693 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 |
This theorem is referenced by: 1lt9 11837 topgrpstr 16655 oppgtset 18474 mgptset 19241 cnfldfun 20551 tngplusg 23245 2logb9irr 25367 hgt750leme 31924 31prm 43753 341fppr2 43892 nfermltl2rev 43901 wtgoldbnnsum4prm 43960 bgoldbnnsum3prm 43962 |
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