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Theorem xrlelttr 12552

Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

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

**Proof of Theorem xrlelttr**
Step	Hyp	Ref	Expression
1		xrleloe 12540	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
2	1	3adant3 1128	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
3		xrlttr 12536	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
4	3	expd 418	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
5		breq1 5071	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))
6	5	biimprd 250	. . . . 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 855	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
9	2, 8	sylbid 242	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)))
10	9	impd 413	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683

This theorem is referenced by: xrletr 12554 xrlelttrd 12556 xrre 12565 xrre2 12566 xrmaxlt 12577 supxrun 12712 iooss1 12776 ico0 12787 iccssioo 12808 iccssico 12811 iocssioo 12830 ioossioo 12832 snunioo 12867 leordtval2 21822 lecldbas 21829 pnfnei 21830 bldisj 23010 xbln0 23026 prdsbl 23103 blsscls2 23116 metcnpi3 23158 iocmnfcld 23379 iscau3 23883 ismbf3d 24257 itgsubst 24648 mdegaddle 24670 mdegmullem 24674 ply1divmo 24731 psercnlem2 25014 ftc1anclem6 34974 ftc1anc 34977 asindmre 34979 snunioo1 41795

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