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Mirrors > Home > ILE 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 3087 | . 2 ⊢ 𝐶 ⊆ 𝐶 | |
2 | xpss12 4616 | . 2 ⊢ ((𝐶 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) | |
3 | 1, 2 | mpan 420 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3041 × cxp 4507 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-in 3047 df-ss 3054 df-opab 3960 df-xp 4515 |
This theorem is referenced by: ssxp2 4946 xpdom3m 6696 axresscn 7636 tx2cn 12366 dvfvalap 12746 |
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