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Mirrors > Home > ILE Home > Th. List > eqlei2 | GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
eqlei2 | ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
2 | eleq1 2180 | . . 3 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
3 | 1, 2 | mpbiri 167 | . 2 ⊢ (𝐵 = 𝐴 → 𝐵 ∈ ℝ) |
4 | eqle 7823 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 = 𝐴) → 𝐵 ≤ 𝐴) | |
5 | 3, 4 | mpancom 418 | 1 ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ℝcr 7587 ≤ cle 7769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: sup3exmid 8683 |
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