xrlenlt - Metamath Proof Explorer

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Theorem xrlenlt 11188

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 5096	. . 3 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
2		opelxpi 5658	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*))
3		df-le 11163	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
4	3	eleq2i 2825	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^*) ∖ ^◡ < ))
5		eldif 3908	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^) ∖ ^◡ < ) ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
6	4, 5	bitri 275	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
7	6	baib 535	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
8	2, 7	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
9	1, 8	bitrid 283	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
10		df-br 5096	. . . 4 ⊢ (𝐵 < 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ < )
11		opelcnvg 5826	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ^◡ < ↔ ⟨𝐵, 𝐴⟩ ∈ < ))
12	10, 11	bitr4id 290	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
13	12	notbid 318	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 𝐵 < 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
14	9, 13	bitr4d 282	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2113 ∖ cdif 3895 ⟨cop 4583 class class class wbr 5095 × cxp 5619 ^◡ccnv 5620 ℝ^*cxr 11156 < clt 11157 ≤ cle 11158

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2712 df-cleq 2725 df-clel 2808 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-br 5096 df-opab 5158 df-xp 5627 df-cnv 5629 df-le 11163

This theorem is referenced by: xrlenltd 11189 xrltnle 11190 lenlt 11202 pnfge 13035 mnfle 13040 xrleloe 13049 xrltlen 13051 xrletri3 13059 xgepnf 13071 xlemnf 13073 xralrple 13111 xleneg 13124 supxr2 13220 supxrbnd1 13227 supxrbnd2 13228 supxrleub 13232 supxrbnd 13234 xrsupssd 13239 infxrgelb 13242 ioon0 13278 iccid 13297 icc0 13300 icoun 13382 ioounsn 13384 snunico 13386 ioodisj 13389 ioojoin 13390 hashgt0elex 14315 hashgt12el 14336 hashgt12el2 14337 0ringnnzr 20449 lecldbas 23154 xmetgt0 24293 icopnfcld 24702 ioombl 25513 vitalilem4 25559 itg2gt0 25708 nmlnogt0 30798 xrlelttric 32760 xrge0infss 32768 joiniooico 32782 xeqlelt 32784 iocinif 32789 esumsnf 34149 esum2d 34178 oms0 34382 omssubadd 34385 cusgracyclt3v 35272 relowlpssretop 37481 mblfinlem3 37772 mblfinlem4 37773 ismblfin 37774 asindmre 37816 dvrelog2b 42232 iocmbl 43370 supxrgere 45494 iccdifprioo 45678 iccpartnel 47600 iccdisj2 49058

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