Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xpss2 | Structured version Visualization version GIF version |
Description: Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.) |
Ref | Expression |
---|---|
xpss2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3986 | . 2 ⊢ 𝐶 ⊆ 𝐶 | |
2 | xpss12 5563 | . 2 ⊢ ((𝐶 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) | |
3 | 1, 2 | mpan 686 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3933 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-in 3940 df-ss 3949 df-opab 5120 df-xp 5554 |
This theorem is referenced by: xpdom3 8603 marypha1lem 8885 canthp1lem2 10063 axresscn 10558 imasvscafn 16798 imasvscaf 16800 gass 18369 gsum2d 19021 tx2cn 22146 txtube 22176 txcmplem1 22177 hausdiag 22181 xkoinjcn 22223 caussi 23827 dvfval 24422 issh2 28913 qtophaus 30999 2ndmbfm 31418 sxbrsigalem0 31428 cvmlift2lem9 32455 cvmlift2lem11 32457 filnetlem3 33625 bj-idres 34344 idresssidinxp 35447 trclexi 39858 cnvtrcl0 39864 ovolval5lem2 42812 ovnovollem1 42815 ovnovollem2 42816 |
Copyright terms: Public domain | W3C validator |