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| Mirrors > Home > MPE Home > Th. List > rexpssxrxp | Structured version Visualization version GIF version | ||
| Description: The Cartesian product of standard reals are a subset of the Cartesian product of extended reals. (Contributed by David A. Wheeler, 8-Dec-2018.) | 
| Ref | Expression | 
|---|---|
| rexpssxrxp | ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ressxr 11305 | . 2 ⊢ ℝ ⊆ ℝ* | |
| 2 | xpss12 5700 | . 2 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) | |
| 3 | 1, 1, 2 | mp2an 692 | 1 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3951 × cxp 5683 ℝcr 11154 ℝ*cxr 11294 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-opab 5206 df-xp 5691 df-xr 11299 | 
| This theorem is referenced by: ltrelxr 11322 xrsdsre 24832 ovolfioo 25502 ovolficc 25503 ovolficcss 25504 ovollb 25514 ovolicc2 25557 ovolfs2 25606 uniiccdif 25613 uniioovol 25614 uniiccvol 25615 uniioombllem2 25618 uniioombllem3a 25619 uniioombllem3 25620 uniioombllem4 25621 uniioombllem5 25622 uniioombl 25624 dyadmbllem 25634 opnmbllem 25636 icoreresf 37353 icoreelrn 37362 relowlpssretop 37365 opnmbllem0 37663 mblfinlem1 37664 mblfinlem2 37665 voliooicof 46011 ovolval3 46662 ovolval4lem2 46665 ovolval5lem2 46668 ovolval5lem3 46669 ovnovollem1 46671 ovnovollem2 46672 | 
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