Theorem lerelxr 11237

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 11214	. 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
2		difss 4099	. 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*)
3	1, 2	eqsstri 3993	1 ⊢ ≤ ⊆ (ℝ* × ℝ*)

Syntax hints:   ∖ cdif 3911   ⊆ wss 3914   × cxp 5636  ^◡ccnv 5637  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-ss 3931  df-le 11214

This theorem is referenced by:  lerel  11238  dfle2  13107  dflt2  13108  ledm  18549  lern  18550  letsr  18552  xrsle  21299  znle  21446  leex  42234  i0oii  48908  io1ii  48909