Theorem lerelxr 11242

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 11219	. 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
2		difss 4089	. 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*)
3	1, 2	eqsstri 3982	1 ⊢ ≤ ⊆ (ℝ* × ℝ*)

Syntax hints:   ∖ cdif 3901   ⊆ wss 3904   × cxp 5643  ^◡ccnv 5644  ℝ^*cxr 11212   < clt 11213   ≤ cle 11214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-ss 3921  df-le 11219

This theorem is referenced by:  lerel  11243  dfle2  13146  dflt2  13147  xrsle  17617  ledm  18605  lern  18606  letsr  18608  znle  21568  leex  42826  i0oii  49505  io1ii  49506