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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 10669 | . 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | |
2 | difss 4105 | . 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*) | |
3 | 1, 2 | eqsstri 3998 | 1 ⊢ ≤ ⊆ (ℝ* × ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3930 ⊆ wss 3933 × cxp 5546 ◡ccnv 5547 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-le 10669 |
This theorem is referenced by: lerel 10693 dfle2 12528 dflt2 12529 ledm 17822 lern 17823 letsr 17825 xrsle 20493 znle 20611 |
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