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Mirrors > Home > MPE Home > Th. List > lerel | Structured version Visualization version GIF version |
Description: "Less than 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 11214 | . 2 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
2 | relxp 5649 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 5735 | . 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ )) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel ≤ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3908 × cxp 5629 Rel wrel 5636 ℝ*cxr 11184 ≤ cle 11186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 df-opab 5166 df-xp 5637 df-rel 5638 df-le 11191 |
This theorem is referenced by: dfle2 13058 dflt2 13059 ledm 18471 lern 18472 lefld 18473 letsr 18474 dvle 25355 gtiso 31498 |
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