xrletr - Intuitionistic Logic Explorer

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Theorem xrletr 10140

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 10128	. . . . . 6 ⊢ < Or ℝ^*
2		sowlin 4440	. . . . . 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 1237	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
5		orcom 736	. . . 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 636	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) → ¬ 𝐶 < 𝐴))
8		xrlenlt 8337	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
9	8	3adant3 1044	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
10		xrlenlt 8337	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
11	10	3adant1 1042	. . . 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 760	. . 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 8337	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴))
16	15	3adant2 1043	. 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 716 ∧ w3a 1005 ∈ wcel 2203 class class class wbr 4108 Or wor 4415 ℝ^*cxr 8306 < clt 8307 ≤ cle 8308

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-cnex 8217 ax-resscn 8218 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2814 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-br 4109 df-opab 4171 df-po 4416 df-iso 4417 df-xp 4754 df-cnv 4756 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313

This theorem is referenced by: xrletrd 10144 xle2add 10211 icc0r 10258 iccss 10273 icossico 10275 iccss2 10276 iccssico 10277 bdxmet 15358

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