Theorem lerelxr 11281

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 11258	. 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
2		difss 4130	. 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*)
3	1, 2	eqsstri 4015	1 ⊢ ≤ ⊆ (ℝ* × ℝ*)

Syntax hints:   ∖ cdif 3944   ⊆ wss 3947   × cxp 5673  ^◡ccnv 5674  ℝ^*cxr 11251   < clt 11252   ≤ cle 11253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-le 11258

This theorem is referenced by:  lerel  11282  dfle2  13130  dflt2  13131  ledm  18547  lern  18548  letsr  18550  xrsle  21165  znle  21307  i0oii  47639  io1ii  47640