Theorem lerelxr 11325

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 11302	. 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
2		difss 4135	. 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*)
3	1, 2	eqsstri 4029	1 ⊢ ≤ ⊆ (ℝ* × ℝ*)

Syntax hints:   ∖ cdif 3947   ⊆ wss 3950   × cxp 5682  ^◡ccnv 5683  ℝ^*cxr 11295   < clt 11296   ≤ cle 11297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953  df-ss 3967  df-le 11302

This theorem is referenced by:  lerel  11326  dfle2  13190  dflt2  13191  ledm  18636  lern  18637  letsr  18639  xrsle  21401  znle  21552  leex  42287  i0oii  48824  io1ii  48825