xrlenlt - Intuitionistic Logic Explorer

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Theorem xrlenlt 8338

Description: 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
xrlenlt	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))

**Proof of Theorem xrlenlt**
Step	Hyp	Ref	Expression
1		df-br 4110	. . 3 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
2		opelxpi 4781	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*))
3		df-le 8314	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
4	3	eleq2i 2299	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^*) ∖ ^◡ < ))
5		eldif 3220	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^) ∖ ^◡ < ) ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
6	4, 5	bitri 184	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
7	6	baib 927	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
8	2, 7	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
9	1, 8	bitrid 192	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
10		df-br 4110	. . . 4 ⊢ (𝐵 < 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ < )
11		opelcnvg 4935	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ^◡ < ↔ ⟨𝐵, 𝐴⟩ ∈ < ))
12	10, 11	bitr4id 199	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
13	12	notbid 673	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 𝐵 < 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
14	9, 13	bitr4d 191	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2203 ∖ cdif 3208 ⟨cop 3692 class class class wbr 4109 × cxp 4747 ^◡ccnv 4748 ℝ^*cxr 8307 < clt 8308 ≤ cle 8309

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-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 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-ral 2525 df-rex 2526 df-v 2815 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-br 4110 df-opab 4172 df-xp 4755 df-cnv 4757 df-le 8314

This theorem is referenced by: lenlt 8349 pnfge 10122 mnfle 10125 xrltle 10131 xrleid 10133 xnn0dcle 10135 xrletri3 10137 xrlelttr 10139 xrltletr 10140 xrletr 10141 xgepnf 10149 xleneg 10170 xltadd1 10209 xsubge0 10214 xleaddadd 10220 iccid 10258 icc0r 10259 icodisj 10325 ioodisj 10326 ioo0 10619 ico0 10621 ioc0 10622 leisorel 11209 xrmaxleim 11929 xrmaxiflemval 11935 xrmaxlesup 11944 xrmaxaddlem 11945 xrminmax 11950 pcadd 13038 bldisj 15266 bdxmet 15366 bdbl 15368

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