xrlenlt - Intuitionistic Logic Explorer

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Theorem xrlenlt 8234

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 4087	. . 3 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
2		opelxpi 4755	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*))
3		df-le 8210	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
4	3	eleq2i 2296	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^*) ∖ ^◡ < ))
5		eldif 3207	. . . . . 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 924	. . . 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 4087	. . . 4 ⊢ (𝐵 < 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ < )
11		opelcnvg 4908	. . . 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 671	. 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 2200 ∖ cdif 3195 ⟨cop 3670 class class class wbr 4086 × cxp 4721 ^◡ccnv 4722 ℝ^*cxr 8203 < clt 8204 ≤ cle 8205

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2802 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-br 4087 df-opab 4149 df-xp 4729 df-cnv 4731 df-le 8210

This theorem is referenced by: lenlt 8245 pnfge 10014 mnfle 10017 xrltle 10023 xrleid 10025 xnn0dcle 10027 xrletri3 10029 xrlelttr 10031 xrltletr 10032 xrletr 10033 xgepnf 10041 xleneg 10062 xltadd1 10101 xsubge0 10106 xleaddadd 10112 iccid 10150 icc0r 10151 icodisj 10217 ioodisj 10218 ioo0 10509 ico0 10511 ioc0 10512 leisorel 11091 xrmaxleim 11795 xrmaxiflemval 11801 xrmaxlesup 11810 xrmaxaddlem 11811 xrminmax 11816 pcadd 12903 bldisj 15115 bdxmet 15215 bdbl 15217

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