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Mirrors > Home > ILE Home > Th. List > xpss1 | GIF version |
Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.) |
Ref | Expression |
---|---|
xpss1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3112 | . 2 ⊢ 𝐶 ⊆ 𝐶 | |
2 | xpss12 4641 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐶) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) | |
3 | 1, 2 | mpan2 421 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3066 × cxp 4532 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-in 3072 df-ss 3079 df-opab 3985 df-xp 4540 |
This theorem is referenced by: ssres2 4841 ssxp1 4970 funssxp 5287 tposssxp 6139 tpostpos2 6155 tfrlemibfn 6218 tfr1onlembfn 6234 tfrcllembfn 6247 enq0enq 7232 tx1cn 12427 |
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