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Mirrors > Home > ILE Home > Th. List > xpss | GIF version |
Description: A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
xpss | ⊢ (𝐴 × 𝐵) ⊆ (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3191 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 3191 | . 2 ⊢ 𝐵 ⊆ V | |
3 | xpss12 4747 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴 × 𝐵) ⊆ (V × V) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2751 ⊆ wss 3143 × cxp 4638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-v 2753 df-in 3149 df-ss 3156 df-opab 4079 df-xp 4646 |
This theorem is referenced by: relxp 4749 eqbrrdva 4811 relrelss 5169 funinsn 5279 eqopi 6190 op1steq 6197 dfoprab4 6210 f1od2 6253 frecuzrdgtcl 10429 frecuzrdgfunlem 10436 reldvdsrsrg 13402 upxp 14155 |
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