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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 2798 | . . . 4 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 863 | . . 3 ⊢ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴) |
3 | leloe 10716 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 ≤ 𝐴 ↔ (𝐴 < 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 261 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ 𝐴) |
5 | 4 | anidms 570 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-pre-lttri 10600 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: eqle 10731 mulge0 11147 msqge0 11150 leidi 11163 leidd 11195 lemulge11 11491 lediv2a 11523 nn2ge 11652 max1ALT 12567 lo1const 14969 isumless 15192 retos 20307 itg2itg1 24340 itg20 24341 nmobndi 28558 breprexp 32014 relowlpssretop 34781 iuneqfzuzlem 41966 fmuldfeq 42225 volioc 42614 caratheodorylem1 43165 |
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