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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 3169 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 3169 | . 2 ⊢ 𝐵 ⊆ V | |
3 | xpss12 4716 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 424 | 1 ⊢ (𝐴 × 𝐵) ⊆ (V × V) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2730 ⊆ wss 3121 × cxp 4607 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-opab 4049 df-xp 4615 |
This theorem is referenced by: relxp 4718 eqbrrdva 4779 relrelss 5135 funinsn 5245 eqopi 6148 op1steq 6155 dfoprab4 6168 f1od2 6211 frecuzrdgtcl 10355 frecuzrdgfunlem 10362 upxp 13025 |
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