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| Mirrors > Home > MPE Home > Th. List > lerelxr | Structured version Visualization version GIF version | ||
| Description: "Less than or equal to" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| lerelxr | ⊢ ≤ ⊆ (ℝ* × ℝ*) | 
| 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*) | 
| Colors of variables: wff setvar class | 
| 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 | 
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