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| Description: A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.) | 
| Ref | Expression | 
|---|---|
| max2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexr 11307 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
| 2 | rexr 11307 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
| 3 | xrmax2 13218 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
| 4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ifcif 4525 class class class wbr 5143 ℝcr 11154 ℝ*cxr 11294 ≤ 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-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-po 5592 df-so 5593 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: lemaxle 13237 z2ge 13240 ssfzunsnext 13609 uzsup 13903 expmulnbnd 14274 discr1 14278 rexuzre 15391 caubnd 15397 limsupgre 15517 limsupbnd2 15519 rlim3 15534 lo1bdd2 15560 o1lo1 15573 rlimclim1 15581 lo1mul 15664 rlimno1 15690 cvgrat 15919 ruclem10 16275 bitsfzo 16472 1arith 16965 evth 24991 ioombl1lem4 25596 itg2monolem3 25787 itgle 25845 ibladdlem 25855 plyaddlem1 26252 coeaddlem 26288 o1cxp 27018 cxp2lim 27020 cxploglim2 27022 ftalem1 27116 ftalem2 27117 chtppilim 27519 dchrisumlem3 27535 ostth2lem2 27678 ostth2lem3 27679 ostth2lem4 27680 ostth3 27682 knoppndvlem18 36530 ibladdnclem 37683 ftc1anclem5 37704 irrapxlem4 42836 irrapxlem5 42837 rexabslelem 45429 uzublem 45441 max2d 45469 climsuse 45623 limsupubuzlem 45727 limsupmnfuzlem 45741 limsupequzmptlem 45743 limsupre3uzlem 45750 liminflelimsuplem 45790 ioodvbdlimc1lem2 45947 ioodvbdlimc2lem 45949 hoidifhspdmvle 46635 | 
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