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Mirrors > Home > NFE Home > Th. List > xpss2 | GIF version |
Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.) |
Ref | Expression |
---|---|
xpss2 | ⊢ (A ⊆ B → (C × A) ⊆ (C × B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3290 | . 2 ⊢ C ⊆ C | |
2 | xpss12 4855 | . 2 ⊢ ((C ⊆ C ∧ A ⊆ B) → (C × A) ⊆ (C × B)) | |
3 | 1, 2 | mpan 651 | 1 ⊢ (A ⊆ B → (C × A) ⊆ (C × B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3257 × cxp 4770 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-xp 4784 |
This theorem is referenced by: ssrnres 5059 fvfullfunlem3 5863 |
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