xrletr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrletr		GIF version

Theorem xrletr 9877

Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

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

**Proof of Theorem xrletr**
Step	Hyp	Ref	Expression
1		xrltso 9865	. . . . . 6 ⊢ < Or ℝ^*
2		sowlin 4352	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*)) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
3	1, 2	mpan 424	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
4	3	3coml 1212	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
5		orcom 729	. . . 4 ⊢ ((𝐶 < 𝐵 ∨ 𝐵 < 𝐴) ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵))
6	4, 5	imbitrdi 161	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐴 → (𝐵 < 𝐴 ∨ 𝐶 < 𝐵)))
7	6	con3d 632	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) → ¬ 𝐶 < 𝐴))
8		xrlenlt 8086	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
9	8	3adant3 1019	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
10		xrlenlt 8086	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
11	10	3adant1 1017	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
12	9, 11	anbi12d 473	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐶 < 𝐵)))
13		ioran 753	. . 3 ⊢ (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐶 < 𝐵))
14	12, 13	bitr4di 198	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ↔ ¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵)))
15		xrlenlt 8086	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴))
16	15	3adant2 1018	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴))
17	7, 14, 16	3imtr4d 203	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∧ w3a 980 ∈ wcel 2164 class class class wbr 4030 Or wor 4327 ℝ^*cxr 8055 < clt 8056 ≤ cle 8057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-po 4328 df-iso 4329 df-xp 4666 df-cnv 4668 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062

This theorem is referenced by: xrletrd 9881 xle2add 9948 icc0r 9995 iccss 10010 icossico 10012 iccss2 10013 iccssico 10014 bdxmet 14680

W3C validator