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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 7777 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | xpss12 4616 | . 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 3041 × cxp 4507 ℝcr 7587 ℝ*cxr 7767 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-opab 3960 df-xp 4515 df-xr 7772 |
This theorem is referenced by: ltrelxr 7793 |
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