Theorem lerel 11198

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 11197	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5641	. 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 3905   × cxp 5621  Rel wrel 5628  ℝ^*cxr 11167   ≤ cle 11169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-dif 3908  df-ss 3922  df-opab 5158  df-xp 5629  df-rel 5630  df-le 11174

This theorem is referenced by:  dfle2  13067  dflt2  13068  ledm  18514  lern  18515  lefld  18516  letsr  18517  dvle  25928  gtiso  32657