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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 7969 | . 2 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
2 | relxp 4718 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 4696 | . 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ )) | |
4 | 1, 2, 3 | mp2 16 | 1 ⊢ Rel ≤ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3121 × cxp 4607 Rel wrel 4614 ℝ*cxr 7940 ≤ cle 7942 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-opab 4049 df-xp 4615 df-rel 4616 df-le 7947 |
This theorem is referenced by: (None) |
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