![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > xpexg | GIF version |
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
xpexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsspw 4589 | . 2 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | unexg 4302 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | |
3 | pwexg 4044 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝐵) ∈ V) | |
4 | pwexg 4044 | . . 3 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∈ V → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V) | |
5 | 2, 3, 4 | 3syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V) |
6 | ssexg 4007 | . 2 ⊢ (((𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V) → (𝐴 × 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 408 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1448 Vcvv 2641 ∪ cun 3019 ⊆ wss 3021 𝒫 cpw 3457 × cxp 4475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-opab 3930 df-xp 4483 |
This theorem is referenced by: xpex 4592 sqxpexg 4593 resiexg 4800 cnvexg 5012 coexg 5019 fex2 5227 fabexg 5246 resfunexgALT 5939 cofunexg 5940 fnexALT 5942 opabex3d 5950 opabex3 5951 oprabexd 5956 ofmresex 5966 mpoexxg 6038 tposexg 6085 erex 6383 pmex 6477 mapex 6478 pmvalg 6483 elpmg 6488 fvdiagfn 6517 ixpexgg 6546 ixpsnf1o 6560 map1 6636 xpdom2 6654 xpdom3m 6657 xpen 6668 mapxpen 6671 xpfi 6747 djuex 6843 djuassen 6977 shftfvalg 10431 climconst2 10899 lmfval 12143 txbasex 12207 txopn 12215 txcn 12225 txrest 12226 blfvalps 12313 dvfvalap 12523 |
Copyright terms: Public domain | W3C validator |