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Mirrors > Home > MPE Home > Th. List > xrlenlt | Structured version Visualization version GIF version |
Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
xrlenlt | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5111 | . . 3 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ ) | |
2 | opelxpi 5675 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ⟨𝐴, 𝐵⟩ ∈ (ℝ* × ℝ*)) | |
3 | df-le 11202 | . . . . . . 7 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | |
4 | 3 | eleq2i 2830 | . . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≤ ↔ ⟨𝐴, 𝐵⟩ ∈ ((ℝ* × ℝ*) ∖ ◡ < )) |
5 | eldif 3925 | . . . . . 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 537 | . . . 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 5111 | . . . 4 ⊢ (𝐵 < 𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ < ) | |
11 | opelcnvg 5841 | . . . 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 205 ∧ wa 397 ∈ wcel 2107 ∖ cdif 3912 ⟨cop 4597 class class class wbr 5110 × cxp 5636 ◡ccnv 5637 ℝ*cxr 11195 < clt 11196 ≤ cle 11197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-cnv 5646 df-le 11202 |
This theorem is referenced by: xrlenltd 11228 xrltnle 11229 lenlt 11240 pnfge 13058 mnfle 13062 xrleloe 13070 xrltlen 13072 xrletri3 13080 xgepnf 13091 xlemnf 13093 xralrple 13131 xleneg 13144 supxr2 13240 supxrbnd1 13247 supxrbnd2 13248 supxrleub 13252 supxrbnd 13254 infxrgelb 13261 ioon0 13297 iccid 13316 icc0 13319 icoun 13399 ioounsn 13401 snunico 13403 ioodisj 13406 ioojoin 13407 hashgt0elex 14308 hashgt12el 14329 hashgt12el2 14330 0ringnnzr 20755 lecldbas 22586 xmetgt0 23727 icopnfcld 24147 ioombl 24945 vitalilem4 24991 itg2gt0 25141 nmlnogt0 29781 xrlelttric 31699 xrsupssd 31706 xrge0infss 31707 joiniooico 31719 xeqlelt 31721 iocinif 31726 esumsnf 32703 esum2d 32732 oms0 32937 omssubadd 32940 cusgracyclt3v 33790 relowlpssretop 35864 mblfinlem3 36146 mblfinlem4 36147 ismblfin 36148 asindmre 36190 dvrelog2b 40552 iocmbl 41576 supxrgere 43641 iccdifprioo 43828 iccpartnel 45704 iccdisj2 47004 |
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