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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 4756 | . 2 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) | |
2 | unexg 4461 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | |
3 | pwexg 4198 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝐵) ∈ V) | |
4 | pwexg 4198 | . . 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 4157 | . 2 ⊢ (((𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ V) → (𝐴 × 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 414 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 ⊆ wss 3144 𝒫 cpw 3590 × cxp 4642 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-opab 4080 df-xp 4650 |
This theorem is referenced by: xpex 4759 sqxpexg 4760 resiexg 4970 cnvexg 5184 coexg 5191 fex2 5403 fabexg 5422 resfunexgALT 6134 cofunexg 6135 fnexALT 6137 funexw 6138 opabex3d 6147 opabex3 6148 oprabexd 6153 ofmresex 6163 mpoexxg 6236 tposexg 6284 erex 6584 pmex 6680 mapex 6681 pmvalg 6686 elpmg 6691 fvdiagfn 6720 ixpexgg 6749 ixpsnf1o 6763 map1 6839 xpdom2 6858 xpdom3m 6861 xpen 6874 mapxpen 6877 xpfi 6959 djuex 7073 djuassen 7247 cc2lem 7296 shftfvalg 10862 climconst2 11334 mulgnngsum 13084 releqgg 13176 eqgex 13177 eqgfval 13178 reldvdsrsrg 13459 dvdsrvald 13460 dvdsrex 13465 aprval 13615 aprap 13619 psrval 13961 psrbasg 13968 psrplusgg 13971 lmfval 14169 txbasex 14234 txopn 14242 txcn 14252 txrest 14253 blfvalps 14362 xmetxp 14484 limccnp2lem 14622 limccnp2cntop 14623 dvfvalap 14627 |
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