Theorem lerel 11039

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 11038	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5607	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5692	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3887   × cxp 5587  Rel wrel 5594  ℝ^*cxr 11008   ≤ cle 11010

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-le 11015

This theorem is referenced by:  dfle2  12881  dflt2  12882  ledm  18308  lern  18309  lefld  18310  letsr  18311  dvle  25171  gtiso  31033