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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 2725 | . . . 4 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 864 | . . 3 ⊢ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴) |
3 | xrleloe 13158 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 257 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐴 ≤ 𝐴) |
5 | 4 | anidms 565 | 1 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ℝ*cxr 11279 < clt 11280 ≤ cle 11281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 |
This theorem is referenced by: xrleidd 13166 xrmax1 13189 xrmax2 13190 xrmin1 13191 xrmin2 13192 xlemul1a 13302 iooid 13387 iccid 13404 icc0 13407 ubioc1 13412 lbico1 13413 lbicc2 13476 ubicc2 13477 snunioc 13492 limsupgord 15452 ledm 18585 lern 18586 letsr 18588 xrsxmet 24769 ismbfd 25612 xraddge02 32608 xrstos 32826 elicc3 35932 xreqle 44838 snunioo1 45035 |
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