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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 7963 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | xpss12 4718 | . 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) | |
3 | 1, 1, 2 | mp2an 424 | 1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3121 × cxp 4609 ℝcr 7773 ℝ*cxr 7953 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-opab 4051 df-xp 4617 df-xr 7958 |
This theorem is referenced by: ltrelxr 7980 |
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