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Mirrors > Home > MPE Home > Th. List > xrleid | Structured version Visualization version GIF version |
Description: 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.) |
Ref | Expression |
---|---|
xrleid | ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . . . 4 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 862 | . . 3 ⊢ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴) |
3 | xrleloe 12529 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 260 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐴 ≤ 𝐴) |
5 | 4 | anidms 569 | 1 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ℝ*cxr 10666 < clt 10667 ≤ cle 10668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 |
This theorem is referenced by: xrleidd 12537 xrmax1 12560 xrmax2 12561 xrmin1 12562 xrmin2 12563 xlemul1a 12673 iooid 12758 iccid 12775 icc0 12778 ubioc1 12782 lbico1 12783 lbicc2 12844 ubicc2 12845 snunioc 12858 limsupgord 14821 ledm 17826 lern 17827 letsr 17829 xrsxmet 23409 ismbfd 24232 xraddge02 30472 xrstos 30659 elicc3 33658 xreqle 41576 snunioo1 41778 |
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