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Mirrors > Home > MPE Home > Th. List > 6lt9 | Structured version Visualization version GIF version |
Description: 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
6lt9 | ⊢ 6 < 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6lt7 12338 | . 2 ⊢ 6 < 7 | |
2 | 7lt9 12352 | . 2 ⊢ 7 < 9 | |
3 | 6re 12242 | . . 3 ⊢ 6 ∈ ℝ | |
4 | 7re 12245 | . . 3 ⊢ 7 ∈ ℝ | |
5 | 9re 12251 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11280 | . 2 ⊢ ((6 < 7 ∧ 7 < 9) → 6 < 9) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ 6 < 9 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5105 < clt 11188 6c6 12211 7c7 12212 9c9 12214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 |
This theorem is referenced by: 5lt9 12354 slotstnscsi 17240 psrvalstr 21316 tngvscaOLD 24006 zlmtsetOLD 32486 wtgoldbnnsum4prm 45965 bgoldbnnsum3prm 45967 |
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