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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 7923 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | xpss12 4695 | . 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 3102 × cxp 4586 ℝcr 7733 ℝ*cxr 7913 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-opab 4028 df-xp 4594 df-xr 7918 |
This theorem is referenced by: ltrelxr 7940 |
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