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

Theorem lerel 10315
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 10314 . 2 ≤ ⊆ (ℝ* × ℝ*)
2 relxp 5284 . 2 Rel (ℝ* × ℝ*)
3 relss 5364 . 2 ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
41, 2, 3mp2 9 1 Rel ≤
Colors of variables: wff setvar class
Syntax hints:  wss 3716   × cxp 5265  Rel wrel 5272  *cxr 10286  cle 10288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-dif 3719  df-in 3723  df-ss 3730  df-opab 4866  df-xp 5273  df-rel 5274  df-le 10293
This theorem is referenced by:  dfle2  12194  dflt2  12195  ledm  17446  lern  17447  lefld  17448  letsr  17449  dvle  23990  gtiso  29809
  Copyright terms: Public domain W3C validator