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Mirrors > Home > MPE Home > Th. List > leloe | Structured version Visualization version GIF version |
Description: 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.) |
Ref | Expression |
---|---|
leloe | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lenlt 9995 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
2 | eqcom 2617 | . . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
3 | 2 | orbi1i 541 | . . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵)) |
4 | orcom 401 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
5 | 3, 4 | bitri 263 | . . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
6 | axlttri 9988 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) | |
7 | 6 | ancoms 468 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) |
8 | 7 | con2bid 343 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴)) |
9 | 5, 8 | syl5rbbr 274 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
10 | 1, 9 | bitrd 267 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-pre-lttri 9889 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: ltle 10005 leltne 10006 lelttr 10007 ltletr 10008 letr 10010 leid 10012 ltlen 10017 leloei 10033 leloed 10059 lemul1 10754 lemul1a 10756 squeeze0 10805 fimaxre 10847 sup3 10859 nn0ge0 11195 nn0sub 11220 elnn0z 11267 xlemul1a 11990 modfzo0difsn 12604 om2uzlti 12611 om2uzlt2i 12612 sqlecan 12833 discr 12863 facdiv 12936 facwordi 12938 resqrex 13839 sqrt2irr 14817 lcmf 15184 efgsfo 17975 efgred 17984 itg2mulc 23320 itgabs 23407 dgrlt 23826 sinq12ge0 24064 sineq0 24077 cxpge0 24229 cxplea 24242 cxple2 24243 cxple2a 24245 cxpcn3lem 24288 cxpcn3 24289 cxpaddlelem 24292 cxpaddle 24293 ang180lem3 24341 atanlogaddlem 24440 rlimcnp2 24493 jensen 24515 amgm 24517 htthlem 27158 hiidge0 27339 staddi 28489 stadd3i 28491 poimirlem28 32607 itgaddnclem2 32639 itgabsnc 32649 pellfund14gap 36469 sineq0ALT 38195 icccncfext 38773 iccpartnel 39976 odz2prm2pw 40013 gboge7 40185 bgoldbtbndlem1 40221 ltnltne 40345 elfzolborelfzop1 42103 |
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