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Mirrors > Home > MPE Home > Th. List > leid | Structured version Visualization version GIF version |
Description: 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
leid | ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 862 | . . 3 ⊢ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴) |
3 | leloe 10721 | . . 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 1533 ∈ wcel 2110 class class class wbr 5058 ℝcr 10530 < clt 10669 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-pre-lttri 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 |
This theorem is referenced by: eqle 10736 mulge0 11152 msqge0 11155 leidi 11168 leidd 11200 lemulge11 11496 lediv2a 11528 nn2ge 11658 max1ALT 12573 lo1const 14971 isumless 15194 retos 20756 itg2itg1 24331 itg20 24332 nmobndi 28546 breprexp 31899 relowlpssretop 34639 iuneqfzuzlem 41595 fmuldfeq 41857 volioc 42250 caratheodorylem1 42802 |
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