Theorem lerel 10970

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 10969	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5598	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5682	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3883   × cxp 5578  Rel wrel 5585  ℝ^*cxr 10939   ≤ cle 10941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-opab 5133  df-xp 5586  df-rel 5587  df-le 10946

This theorem is referenced by:  dfle2  12810  dflt2  12811  ledm  18223  lern  18224  lefld  18225  letsr  18226  dvle  25076  gtiso  30935