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Mirrors > Home > ILE Home > Th. List > lerel | GIF version |
Description: 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
lerel | ⊢ Rel ≤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lerelxr 7795 | . 2 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
2 | relxp 4618 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 4596 | . 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ )) | |
4 | 1, 2, 3 | mp2 16 | 1 ⊢ Rel ≤ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3041 × cxp 4507 Rel wrel 4514 ℝ*cxr 7767 ≤ 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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-opab 3960 df-xp 4515 df-rel 4516 df-le 7774 |
This theorem is referenced by: (None) |
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