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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 7175 | . 2 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
2 | relxp 4465 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 4445 | . 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ )) | |
4 | 1, 2, 3 | mp2 16 | 1 ⊢ Rel ≤ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 2973 × cxp 4361 Rel wrel 4368 ℝ*cxr 7152 ≤ cle 7154 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-opab 3840 df-xp 4369 df-rel 4370 df-le 7159 |
This theorem is referenced by: (None) |
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