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Theorem lelttr 11334

Description: Transitive law. (Contributed by NM, 23-May-1999.)

**Assertion**
Ref	Expression
lelttr	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

**Proof of Theorem lelttr**
Step	Hyp	Ref	Expression
1		leloe 11330	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
2	1	3adant3 1130	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
3		lttr 11320	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
4	3	expd 415	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
5		breq1 5151	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))
6	5	biimprd 247	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))
7	6	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 = 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
8	4, 7	jaod 858	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
9	2, 8	sylbid 239	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
10	9	impd 410	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ℝcr 11137 < clt 11278 ≤ cle 11279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284

This theorem is referenced by: leltletr 11335 letr 11338 lelttri 11371 lelttrd 11402 letrp1 12088 ltmul12a 12100 ledivp1 12146 supmul1 12213 bndndx 12501 uzind 12684 fnn0ind 12691 rpnnen1lem5 12995 xrinfmsslem 13319 elfzo0z 13706 nn0p1elfzo 13707 fzofzim 13711 elfzodifsumelfzo 13730 flge 13802 flflp1 13804 flltdivnn0lt 13830 modfzo0difsn 13940 fsequb 13972 expnlbnd2 14228 ccat2s1fvw 14620 swrdswrd 14687 pfxccatin12lem3 14714 repswswrd 14766 caubnd2 15336 caubnd 15337 mulcn2 15572 cn1lem 15574 rlimo1 15593 o1rlimmul 15595 climsqz 15617 climsqz2 15618 rlimsqzlem 15627 climsup 15648 caucvgrlem2 15653 iseralt 15663 cvgcmp 15794 cvgcmpce 15796 ruclem3 16209 ruclem12 16217 ltoddhalfle 16337 algcvgblem 16547 ncoprmlnprm 16699 pclem 16806 infpn2 16881 gsummoncoe1 22226 mp2pm2mplem4 22710 metss2lem 24419 ngptgp 24544 nghmcn 24661 iocopnst 24863 ovollb2lem 25416 ovolicc2lem4 25448 volcn 25534 ismbf3d 25582 dvcnvrelem1 25949 dvfsumrlim 25965 ulmcn 26334 mtest 26339 logdivlti 26553 isosctrlem1 26749 ftalem2 27005 chtub 27144 bposlem6 27221 gausslemma2dlem2 27299 chtppilim 27407 dchrisumlem3 27423 pntlem3 27541 clwlkclwwlklem2a 29807 vacn 30503 nmcvcn 30504 blocni 30614 chscllem2 31447 lnconi 31842 staddi 32055 stadd3i 32057 ltflcei 37081 poimirlem29 37122 geomcau 37232 heibor1lem 37282 bfplem2 37296 rrncmslem 37305 climinf 44994 zm1nn 46682 iccpartigtl 46763 tgoldbach 47157 ply1mulgsumlem2 47455 difmodm1lt 47595

	
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