Theorem lerel 11173

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 11172	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5634	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5722	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3902   × cxp 5614  Rel wrel 5621  ℝ^*cxr 11142   ≤ cle 11144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3905  df-ss 3919  df-opab 5154  df-xp 5622  df-rel 5623  df-le 11149

This theorem is referenced by:  dfle2  13043  dflt2  13044  ledm  18493  lern  18494  lefld  18495  letsr  18496  dvle  25937  gtiso  32677