Theorem lerel 10707

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 10706	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5575	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5658	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3938   × cxp 5555  Rel wrel 5562  ℝ^*cxr 10676   ≤ cle 10678

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-opab 5131  df-xp 5563  df-rel 5564  df-le 10683

This theorem is referenced by:  dfle2  12543  dflt2  12544  ledm  17836  lern  17837  lefld  17838  letsr  17839  dvle  24606  gtiso  30438