MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lerel Structured version   Visualization version   GIF version

Theorem lerel 10046
Description: 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
lerel Rel ≤

Proof of Theorem lerel
StepHypRef Expression
1 lerelxr 10045 . 2 ≤ ⊆ (ℝ* × ℝ*)
2 relxp 5188 . 2 Rel (ℝ* × ℝ*)
3 relss 5167 . 2 ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
41, 2, 3mp2 9 1 Rel ≤
Colors of variables: wff setvar class
Syntax hints:  wss 3555   × cxp 5072  Rel wrel 5079  *cxr 10017  cle 10019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-opab 4674  df-xp 5080  df-rel 5081  df-le 10024
This theorem is referenced by:  dfle2  11924  dflt2  11925  ledm  17145  lern  17146  lefld  17147  letsr  17148  dvle  23674  gtiso  29321
  Copyright terms: Public domain W3C validator