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Mirrors > Home > MPE Home > Th. List > 6lt8 | Structured version Visualization version GIF version |
Description: 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
6lt8 | ⊢ 6 < 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6lt7 11660 | . 2 ⊢ 6 < 7 | |
2 | 7lt8 11666 | . 2 ⊢ 7 < 8 | |
3 | 6re 11564 | . . 3 ⊢ 6 ∈ ℝ | |
4 | 7re 11567 | . . 3 ⊢ 7 ∈ ℝ | |
5 | 8re 11570 | . . 3 ⊢ 8 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10602 | . 2 ⊢ ((6 < 7 ∧ 7 < 8) → 6 < 8) |
7 | 1, 2, 6 | mp2an 688 | 1 ⊢ 6 < 8 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4956 < clt 10510 6c6 11533 7c7 11534 8c8 11535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 |
This theorem is referenced by: 5lt8 11668 631prm 16277 ipsstr 16460 phlstr 16470 sralem 19627 sravsca 19632 chtub 25458 bpos1 25529 |
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