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Mirrors > Home > MPE Home > Th. List > xrleloe | Structured version Visualization version GIF version |
Description: 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
xrleloe | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlenlt 11221 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
2 | xrlttri 13059 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) | |
3 | 2 | ancoms 460 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) |
4 | 3 | con2bid 355 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴)) |
5 | eqcom 2744 | . . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
6 | 5 | orbi1i 913 | . . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵)) |
7 | orcom 869 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
8 | 6, 7 | bitri 275 | . . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
9 | 4, 8 | bitr3di 286 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
10 | 1, 9 | bitrd 279 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 |
This theorem is referenced by: xrleltne 13065 dfle2 13067 xrltle 13069 xrleid 13071 xrlelttr 13076 xrltletr 13077 xrletr 13078 nltpnft 13084 ngtmnft 13086 xmulge0 13204 xlemul1a 13208 xadddi2 13217 prunioo 13399 xrsxmet 24175 metds0 24216 metdseq0 24220 metnrmlem1a 24224 icombl 24931 ioombl 24932 volivth 24974 vitalilem4 24978 itg2gt0 25128 deg1sublt 25478 xrge0addgt0 31885 xrge0adddir 31886 icorempo 35825 icceuelpartlem 45634 |
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