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Mirrors > Home > ILE Home > Th. List > rexpssxrxp | GIF version |
Description: The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
rexpssxrxp | ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressxr 7809 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | xpss12 4646 | . 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) | |
3 | 1, 1, 2 | mp2an 422 | 1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3071 × cxp 4537 ℝcr 7619 ℝ*cxr 7799 |
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-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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-opab 3990 df-xp 4545 df-xr 7804 |
This theorem is referenced by: ltrelxr 7825 |
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