Theorem lerel 11197

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 11196	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5640	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5729	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3890   × cxp 5620  Rel wrel 5627  ℝ^*cxr 11166   ≤ cle 11168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-dif 3893  df-ss 3907  df-opab 5149  df-xp 5628  df-rel 5629  df-le 11173

This theorem is referenced by:  dfle2  13062  dflt2  13063  ledm  18514  lern  18515  lefld  18516  letsr  18517  dvle  25953  gtiso  32763