Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2965 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 2965 | . 2 ⊢ 𝐵 ⊆ V | |
3 | xpss12 4445 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 402 | 1 ⊢ (𝐴 × 𝐵) ⊆ (V × V) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2557 ⊆ 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-v 2559 df-in 2924 df-ss 2931 df-opab 3819 df-xp 4351 |
This theorem is referenced by: relxp 4447 eqbrrdva 4505 relrelss 4844 eqopi 5798 op1steq 5805 dfoprab4 5818 frecuzrdgfn 9198 |
Copyright terms: Public domain | W3C validator |