Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lerelxr | Structured version Visualization version GIF version |
Description: 'Less than or equal' 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 9959 | . 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | |
2 | difss 3699 | . 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*) | |
3 | 1, 2 | eqsstri 3598 | 1 ⊢ ≤ ⊆ (ℝ* × ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3537 ⊆ wss 3540 × cxp 5036 ◡ccnv 5037 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-le 9959 |
This theorem is referenced by: lerel 9981 dfle2 11856 dflt2 11857 ledm 17047 lern 17048 letsr 17050 xrsle 19585 znle 19703 |
Copyright terms: Public domain | W3C validator |