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Mirrors > Home > ILE Home > Th. List > leid | 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 | ltnr 7761 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | lenlt 7760 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐴)) | |
3 | 2 | anidms 392 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | mpbird 166 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 1463 class class class wbr 3895 ℝcr 7543 < clt 7721 ≤ cle 7722 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7633 ax-resscn 7634 ax-pre-ltirr 7654 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-cnv 4507 df-pnf 7723 df-mnf 7724 df-xr 7725 df-ltxr 7726 df-le 7727 |
This theorem is referenced by: eqle 7775 leidi 8163 leidd 8192 lemulge11 8531 lediv2a 8560 |
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