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Mirrors > Home > MPE Home > Th. List > xpex | Structured version Visualization version GIF version |
Description: The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
xpex.1 | ⊢ 𝐴 ∈ V |
xpex.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
xpex | ⊢ (𝐴 × 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | xpex.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | xpexg 6858 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐴 × 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: oprabex 7047 oprabex3 7048 fnpm 7752 mapsnf1o2 7791 ixpsnf1o 7834 xpsnen 7929 endisj 7932 xpcomen 7936 xpassen 7939 xpmapenlem 8012 mapunen 8014 unxpdomlem3 8051 hartogslem1 8330 rankxpl 8621 rankfu 8623 rankmapu 8624 rankxplim 8625 rankxplim2 8626 rankxplim3 8627 rankxpsuc 8628 r0weon 8718 infxpenlem 8719 infxpenc2lem2 8726 dfac3 8827 dfac5lem2 8830 dfac5lem3 8831 dfac5lem4 8832 cdafn 8874 unctb 8910 axcc2lem 9141 axdc3lem 9155 axdc4lem 9160 enqex 9623 nqex 9624 enrex 9767 axcnex 9847 zexALT 11273 cnexALT 11704 addex 11706 mulex 11707 ixxex 12057 shftfval 13658 climconst2 14127 cpnnen 14797 ruclem13 14810 cnso 14815 prdsval 15938 prdsplusg 15941 prdsmulr 15942 prdsvsca 15943 prdsle 15945 prdsds 15947 prdshom 15950 prdsco 15951 fnmrc 16090 mrcfval 16091 mreacs 16142 comfffval 16181 oppccofval 16199 sectfval 16234 brssc 16297 sscpwex 16298 isssc 16303 isfunc 16347 isfuncd 16348 idfu2nd 16360 idfu1st 16362 idfucl 16364 wunfunc 16382 fuccofval 16442 homafval 16502 homaf 16503 homaval 16504 coapm 16544 catccofval 16573 catcfuccl 16582 xpcval 16640 xpcbas 16641 xpchom 16643 xpccofval 16645 1stfval 16654 2ndfval 16657 1stfcl 16660 2ndfcl 16661 catcxpccl 16670 evlf2 16681 evlf1 16683 evlfcl 16685 hof1fval 16716 hof2fval 16718 hofcl 16722 ipoval 16977 letsr 17050 plusffval 17070 frmdplusg 17214 eqgfval 17465 efglem 17952 efgval 17953 scaffval 18704 psrplusg 19202 ltbval 19292 opsrle 19296 evlslem2 19333 evlssca 19343 mpfind 19357 evls1sca 19509 pf1ind 19540 cnfldds 19577 xrsadd 19582 xrsmul 19583 xrsle 19585 xrsds 19608 znle 19703 ipffval 19812 pjfval 19869 mat1dimmul 20101 2ndcctbss 21068 txuni2 21178 txbas 21180 eltx 21181 txcnp 21233 txcnmpt 21237 txrest 21244 txlm 21261 tx1stc 21263 tx2ndc 21264 txkgen 21265 txflf 21620 cnextfval 21676 distgp 21713 indistgp 21714 ustfn 21815 ustn0 21834 ussid 21874 ressuss 21877 ishtpy 22579 isphtpc 22601 elovolm 23050 elovolmr 23051 ovolmge0 23052 ovolgelb 23055 ovolunlem1a 23071 ovolunlem1 23072 ovoliunlem1 23077 ovoliunlem2 23078 ovolshftlem2 23085 ovolicc2 23097 ioombl1 23137 dyadmbl 23174 vitali 23188 mbfimaopnlem 23228 dvfval 23467 plyeq0lem 23770 taylfval 23917 ulmval 23938 dmarea 24484 dchrplusg 24772 iscgrg 25207 ishlg 25297 ishpg 25451 iscgra 25501 isinag 25529 axlowdimlem15 25636 axlowdim 25641 iseupa 26492 isgrpoi 26736 sspval 26962 0ofval 27026 ajfval 27048 hvmulex 27252 inftmrel 29065 isinftm 29066 smatrcl 29190 tpr2rico 29286 faeval 29636 mbfmco2 29654 br2base 29658 sxbrsigalem0 29660 sxbrsigalem3 29661 dya2iocrfn 29668 dya2iocct 29669 dya2iocnrect 29670 dya2iocuni 29672 dya2iocucvr 29673 sxbrsigalem2 29675 eulerpartlemgs2 29769 ccatmulgnn0dir 29945 afsval 30002 cvmlift2lem9 30547 mexval 30653 mdvval 30655 mpstval 30686 brimg 31214 brrestrict 31226 colinearex 31337 poimirlem4 32583 poimirlem28 32607 mblfinlem1 32616 heiborlem3 32782 rrnval 32796 ismrer1 32807 lcvfbr 33325 cmtfvalN 33515 cvrfval 33573 dvhvbase 35394 dvhfvadd 35398 dvhfvsca 35407 dibval 35449 dibfna 35461 dicval 35483 hdmap1fval 36104 mzpincl 36315 pellexlem3 36413 pellexlem4 36414 pellexlem5 36415 aomclem6 36647 trclexi 36946 rtrclexi 36947 brtrclfv2 37038 hoiprodcl2 39445 hoicvrrex 39446 ovn0lem 39455 ovnhoilem1 39491 ovnlecvr2 39500 opnvonmbllem1 39522 opnvonmbllem2 39523 ovolval2lem 39533 ovolval2 39534 ovolval3 39537 ovolval4lem2 39540 ovolval5lem2 39543 ovnovollem1 39546 ovnovollem2 39547 smflimlem6 39662 elpglem3 42255 aacllem 42356 |
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