Theorem lerel 9981

Description: 'Less 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 9980	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5150	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5129	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3540   × cxp 5036  Rel wrel 5043  ℝ^*cxr 9952   ≤ cle 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-opab 4644  df-xp 5044  df-rel 5045  df-le 9959

This theorem is referenced by:  dfle2  11856  dflt2  11857  ledm  17047  lern  17048  lefld  17049  letsr  17050  dvle  23574  gtiso  28861