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Mirrors > Home > MPE Home > Th. List > ltleletr | Structured version Visualization version GIF version |
Description: Transitive law, weaker form of ltletr 10585. (Contributed by AV, 14-Oct-2018.) |
Ref | Expression |
---|---|
ltleletr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpb 1142 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ)) | |
2 | ltletr 10585 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | |
3 | ltle 10582 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶)) | |
4 | 1, 2, 3 | sylsyld 61 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 ∈ wcel 2083 class class class wbr 4968 ℝcr 10389 < clt 10528 ≤ cle 10529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-pre-lttri 10464 ax-pre-lttrn 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 |
This theorem is referenced by: nn0ge2m1nn 11818 lbzbi 12189 hashge2el2dif 13688 wrdlenge2n0 13754 swrdnnn0nd 13858 pfx2 14149 prmgaplem6 16225 chfacfscmul0 21154 chfacfpmmul0 21158 icoopnst 23230 gausslemma2dlem3 25630 gausslemma2dlem4 25631 ltflcei 34432 nnsum4primesevenALTV 43470 m1modmmod 44084 |
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