Theorem lerel 11183

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 11182	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5637	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5726	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3898   × cxp 5617  Rel wrel 5624  ℝ^*cxr 11152   ≤ cle 11154

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 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-ss 3915  df-opab 5156  df-xp 5625  df-rel 5626  df-le 11159

This theorem is referenced by:  dfle2  13048  dflt2  13049  ledm  18498  lern  18499  lefld  18500  letsr  18501  dvle  25940  gtiso  32686