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| Mirrors > Home > MPE Home > Th. List > leloed | Structured version Visualization version GIF version | ||
| Description: 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| Ref | Expression |
|---|---|
| leloed | ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | ltd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 3 | leloe 11347 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: mulge0 11781 prodgt0 12114 lemul1 12119 fimaxre 12212 fiminre 12215 supfirege 12255 nn0le2is012 12682 nn0o1gt2 16418 2mulprm 16730 reconnlem1 24848 reconnlem2 24849 ivthle 25491 ivthle2 25492 ovolicc2lem3 25554 itgsplitioo 25873 dvlip 26032 dvge0 26045 dvfsumlem1 26066 dgrco 26315 plydivex 26339 coseq00topi 26544 logreclem 26805 scvxcvx 27029 pntrlog2bndlem5 27625 fzo0opth 32807 dnibndlem13 36491 lcmineqlem23 42052 lcmineqlem 42053 aks4d1p1 42077 sticksstones12a 42158 sticksstones22 42169 metakunt9 42214 elpell1qr2 42883 pellfundex 42897 fmul01lt1lem2 45600 wallispilem3 46082 fourierdlem25 46147 fourierdlem42 46164 lighneallem4b 47596 nn0o1gt2ALTV 47681 stgoldbwt 47763 sbgoldbwt 47764 sbgoldbalt 47768 nnsum3primesle9 47781 bgoldbtbndlem1 47792 |
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