xrlenlt - Metamath Proof Explorer

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Theorem xrlenlt 11317

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 5153	. . 3 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
2		opelxpi 5719	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*))
3		df-le 11292	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
4	3	eleq2i 2821	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^*) ∖ ^◡ < ))
5		eldif 3959	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((ℝ^* × ℝ^) ∖ ^◡ < ) ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^ × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
6	4, 5	bitri 274	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ (⟨𝐴, 𝐵⟩ ∈ (ℝ^* × ℝ^*) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
7	6	baib 534	. . . 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 282	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
10		df-br 5153	. . . 4 ⊢ (𝐵 < 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ < )
11		opelcnvg 5887	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (⟨𝐴, 𝐵⟩ ∈ ^◡ < ↔ ⟨𝐵, 𝐴⟩ ∈ < ))
12	10, 11	bitr4id 289	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
13	12	notbid 317	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 𝐵 < 𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ^◡ < ))
14	9, 13	bitr4d 281	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∖ cdif 3946 ⟨cop 4638 class class class wbr 5152 × cxp 5680 ^◡ccnv 5681 ℝ^*cxr 11285 < clt 11286 ≤ cle 11287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-le 11292

This theorem is referenced by: xrlenltd 11318 xrltnle 11319 lenlt 11330 pnfge 13150 mnfle 13154 xrleloe 13163 xrltlen 13165 xrletri3 13173 xgepnf 13184 xlemnf 13186 xralrple 13224 xleneg 13237 supxr2 13333 supxrbnd1 13340 supxrbnd2 13341 supxrleub 13345 supxrbnd 13347 infxrgelb 13354 ioon0 13390 iccid 13409 icc0 13412 icoun 13492 ioounsn 13494 snunico 13496 ioodisj 13499 ioojoin 13500 hashgt0elex 14400 hashgt12el 14421 hashgt12el2 14422 0ringnnzr 20469 lecldbas 23143 xmetgt0 24284 icopnfcld 24704 ioombl 25514 vitalilem4 25560 itg2gt0 25710 nmlnogt0 30627 xrlelttric 32543 xrsupssd 32550 xrge0infss 32551 joiniooico 32563 xeqlelt 32565 iocinif 32570 esumsnf 33716 esum2d 33745 oms0 33950 omssubadd 33953 cusgracyclt3v 34799 relowlpssretop 36876 mblfinlem3 37165 mblfinlem4 37166 ismblfin 37167 asindmre 37209 dvrelog2b 41569 iocmbl 42672 supxrgere 44744 iccdifprioo 44930 iccpartnel 46807 iccdisj2 47994

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