Theorem lerel 11278

Description: "Less than 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

Step	Hyp	Ref	Expression
1		lerelxr 11277	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5695	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5782	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3949   × cxp 5675  Rel wrel 5682  ℝ^*cxr 11247   ≤ cle 11249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-le 11254

This theorem is referenced by:  dfle2  13126  dflt2  13127  ledm  18543  lern  18544  lefld  18545  letsr  18546  dvle  25524  gtiso  31922