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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 3046 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 3046 | . 2 ⊢ 𝐵 ⊆ V | |
3 | xpss12 4545 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 417 | 1 ⊢ (𝐴 × 𝐵) ⊆ (V × V) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2619 ⊆ wss 2999 × cxp 4436 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-in 3005 df-ss 3012 df-opab 3900 df-xp 4444 |
This theorem is referenced by: relxp 4547 eqbrrdva 4606 relrelss 4957 funinsn 5063 eqopi 5942 op1steq 5949 dfoprab4 5962 f1od2 6000 frecuzrdgtcl 9819 frecuzrdgfunlem 9826 |
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