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Mirrors > Home > ILE Home > Th. List > lerelxr | 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 7799 | . 2 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | |
2 | difss 3197 | . 2 ⊢ ((ℝ* × ℝ*) ∖ ◡ < ) ⊆ (ℝ* × ℝ*) | |
3 | 1, 2 | eqsstri 3124 | 1 ⊢ ≤ ⊆ (ℝ* × ℝ*) |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 3063 ⊆ wss 3066 × cxp 4532 ◡ccnv 4533 ℝ*cxr 7792 < clt 7793 ≤ cle 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-le 7799 |
This theorem is referenced by: lerel 7821 |
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