![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opelxpd | Structured version Visualization version GIF version |
Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
opelxpd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
opelxpd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
Ref | Expression |
---|---|
opelxpd | ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | opelxpd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
3 | opelxpi 5714 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)) | |
4 | 1, 2, 3 | syl2anc 585 | 1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⟨cop 4635 × cxp 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 df-xp 5683 |
This theorem is referenced by: otel3xp 5723 opabssxpd 5724 relssdmrn 6268 fpr2g 7213 fliftrel 7305 elovimad 7457 el2xptp0 8022 oprab2co 8083 1stconst 8086 2ndconst 8087 curry2 8093 fsplitfpar 8104 offsplitfpar 8105 fnwelem 8117 xpf1o 9139 xpmapenlem 9144 unxpdomlem3 9252 fseqenlem1 10019 fseqenlem2 10020 iundom2g 10535 ordpipq 10937 addpqf 10939 mulpqf 10941 recmulnq 10959 ltexnq 10970 axmulf 11141 cnrecnv 15112 ruclem1 16174 eucalgf 16520 qredeu 16595 crth 16711 phimullem 16712 prmreclem3 16851 setsstruct2 17107 imasaddflem 17476 xpsaddlem 17519 xpsvsca 17523 xpsle 17525 comffval 17643 oppccofval 17661 isoval 17712 brcic 17745 funcf2 17818 idfu2nd 17827 resf2nd 17845 wunfunc 17849 wunfuncOLD 17850 homaval 17981 setcco 18033 catcco 18055 estrcco 18081 xpcco 18135 xpchom2 18138 xpcco2 18139 xpccatid 18140 prfcl 18155 prf1st 18156 prf2nd 18157 evlf2 18171 curf1cl 18181 curf2cl 18184 curfcl 18185 uncf1 18189 uncf2 18190 uncfcurf 18192 diag11 18196 diag12 18197 diag2 18198 curf2ndf 18200 hof2fval 18208 yonedalem21 18226 yonedalem22 18231 yonedalem3b 18232 yonffthlem 18235 latcl2 18389 xpsmnd0 18666 xpsinv 18943 xpsgrpsub 18944 lsmhash 19573 frgpuplem 19640 xpsring1d 20146 mdetrlin 22104 mdetrsca 22105 txcls 23108 txcnp 23124 txcnmpt 23128 txdis1cn 23139 txlly 23140 txnlly 23141 txlm 23152 lmcn2 23153 txkgen 23156 xkococnlem 23163 txhmeo 23307 ptuncnv 23311 txflf 23510 flfcnp2 23511 tmdcn2 23593 qustgplem 23625 tsmsadd 23651 imasdsf1olem 23879 xpsdsval 23887 comet 24022 metustid 24063 metustexhalf 24065 metuel2 24074 tngnm 24168 cnheiborlem 24470 bcthlem5 24845 ovollb2lem 25005 ovolctb 25007 ovoliunlem2 25020 ovolshftlem1 25026 ovolscalem1 25030 ovolicc1 25033 ioombl1lem1 25075 dyadf 25108 itg1addlem4 25216 itg1addlem4OLD 25217 limccnp2 25409 dvaddbr 25455 dvmulbr 25456 dvcobr 25463 lhop1lem 25530 cxpcn3 26256 dvdsmulf1o 26698 addsqnreup 26946 addsval 27446 mulsval 27565 tgjustc1 27726 tgjustc2 27727 tgelrnln 27881 numclwwlk1lem2f 29608 ofresid 31867 fsuppcurry1 31950 fsuppcurry2 31951 gsumpart 32207 prsdm 32894 prsrn 32895 esum2dlem 33090 hgt750lemb 33668 cvmlift2lem10 34303 goelel3xp 34339 sat1el2xp 34370 fmla0xp 34374 prv1n 34422 pprodss4v 34856 gg-dvmulbr 35175 gg-dvcobr 35176 poimirlem3 36491 poimirlem4 36492 poimirlem17 36505 poimirlem20 36508 mblfinlem2 36526 f1o2d2 41055 projf1o 43896 ovolval4lem1 45365 ovolval5lem2 45369 rngqiprngimf 46782 pzriprnglem4 46808 pzriprnglem5 46809 pzriprnglem8 46812 pzriprnglem12 46816 |
Copyright terms: Public domain | W3C validator |