Theorem lerelxr 11199

Description: "Less than or equal to" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
lerelxr	⊢ ≤ ⊆ (ℝ* × ℝ*)

Proof of Theorem lerelxr

Step	Hyp	Ref	Expression
1		df-le 11176	. 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
2		difss 4077	. 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*)
3	1, 2	eqsstri 3969	1 ⊢ ≤ ⊆ (ℝ* × ℝ*)

Syntax hints:   ∖ cdif 3887   ⊆ wss 3890   × cxp 5622  ^◡ccnv 5623  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171

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-le 11176

This theorem is referenced by:  lerel  11200  dfle2  13089  dflt2  13090  xrsle  17559  ledm  18547  lern  18548  letsr  18550  znle  21526  leex  42699  i0oii  49407  io1ii  49408