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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 7833 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | xpss12 4654 | . 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) | |
3 | 1, 1, 2 | mp2an 423 | 1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3076 × cxp 4545 ℝcr 7643 ℝ*cxr 7823 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-opab 3998 df-xp 4553 df-xr 7828 |
This theorem is referenced by: ltrelxr 7849 |
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