Theorem lerel 10315

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 10314	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5284	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5364	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3716   × cxp 5265  Rel wrel 5272  ℝ^*cxr 10286   ≤ cle 10288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-dif 3719  df-in 3723  df-ss 3730  df-opab 4866  df-xp 5273  df-rel 5274  df-le 10293

This theorem is referenced by:  dfle2  12194  dflt2  12195  ledm  17446  lern  17447  lefld  17448  letsr  17449  dvle  23990  gtiso  29809