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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 2964 | . 2 ⊢ 𝐶 ⊆ 𝐶 | |
2 | xpss12 4445 | . 2 ⊢ ((𝐶 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) | |
3 | 1, 2 | mpan 400 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 2917 × cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-in 2924 df-ss 2931 df-opab 3819 df-xp 4351 |
This theorem is referenced by: ssxp2 4758 xpdom3m 6308 axresscn 6936 |
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