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Mirrors > Home > ILE Home > Th. List > opelxpi | GIF version |
Description: Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
opelxpi | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4539 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | biimpri 132 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 〈cop 3500 × cxp 4507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 |
This theorem is referenced by: opelxpd 4542 opelvvg 4558 opelvv 4559 opbrop 4588 fliftrel 5661 fnotovb 5782 ovi3 5875 ovres 5878 fovrn 5881 fnovrn 5886 ovconst2 5890 oprab2co 6083 1stconst 6086 2ndconst 6087 f1od2 6100 brdifun 6424 ecopqsi 6452 brecop 6487 th3q 6502 xpcomco 6688 xpf1o 6706 xpmapenlem 6711 djulclr 6902 djurclr 6903 djulcl 6904 djurcl 6905 djuf1olem 6906 addpiord 7092 mulpiord 7093 enqeceq 7135 1nq 7142 addpipqqslem 7145 mulpipq 7148 mulpipqqs 7149 addclnq 7151 mulclnq 7152 recexnq 7166 ltexnqq 7184 prarloclemarch 7194 prarloclemarch2 7195 nnnq 7198 enq0breq 7212 enq0eceq 7213 nqnq0 7217 addnnnq0 7225 mulnnnq0 7226 addclnq0 7227 mulclnq0 7228 nqpnq0nq 7229 prarloclemlt 7269 prarloclemlo 7270 prarloclemcalc 7278 genpelxp 7287 nqprm 7318 ltexprlempr 7384 recexprlempr 7408 cauappcvgprlemcl 7429 cauappcvgprlemladd 7434 caucvgprlemcl 7452 caucvgprprlemcl 7480 enreceq 7512 addsrpr 7521 mulsrpr 7522 0r 7526 1sr 7527 m1r 7528 addclsr 7529 mulclsr 7530 prsrcl 7560 mappsrprg 7580 addcnsr 7610 mulcnsr 7611 addcnsrec 7618 mulcnsrec 7619 pitonnlem2 7623 pitonn 7624 pitore 7626 recnnre 7627 axaddcl 7640 axmulcl 7642 xrlenlt 7797 frecuzrdgg 10157 frecuzrdgsuctlem 10164 seq3val 10199 cnrecnv 10650 eucalgf 11663 eucalg 11667 qredeu 11705 qnumdenbi 11797 crth 11827 phimullem 11828 setscom 11926 setsslid 11936 txbas 12354 upxp 12368 uptx 12370 txlm 12375 cnmpt21 12387 txswaphmeolem 12416 txswaphmeo 12417 comet 12595 qtopbasss 12617 cnmetdval 12625 remetdval 12635 tgqioo 12643 dvcnp2cntop 12759 dvef 12783 djucllem 12934 pwle2 13120 |
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