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Mirrors > Home > ILE Home > Th. List > opelxpi | Unicode version |
Description: Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
opelxpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4392 | . 2 | |
2 | 1 | biimpri 131 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wcel 1433 cop 3401 cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 |
This theorem is referenced by: opelvvg 4407 opelvv 4408 opbrop 4437 fliftrel 5452 fnotovb 5568 ovi3 5657 ovres 5660 fovrn 5663 fnovrn 5668 ovconst2 5672 oprab2co 5859 1stconst 5862 2ndconst 5863 f1od2 5876 brdifun 6156 ecopqsi 6184 brecop 6219 th3q 6234 xpcomco 6323 addpiord 6506 mulpiord 6507 enqeceq 6549 1nq 6556 addpipqqslem 6559 mulpipq 6562 mulpipqqs 6563 addclnq 6565 mulclnq 6566 recexnq 6580 ltexnqq 6598 prarloclemarch 6608 prarloclemarch2 6609 nnnq 6612 enq0breq 6626 enq0eceq 6627 nqnq0 6631 addnnnq0 6639 mulnnnq0 6640 addclnq0 6641 mulclnq0 6642 nqpnq0nq 6643 prarloclemlt 6683 prarloclemlo 6684 prarloclemcalc 6692 genpelxp 6701 nqprm 6732 ltexprlempr 6798 recexprlempr 6822 cauappcvgprlemcl 6843 cauappcvgprlemladd 6848 caucvgprlemcl 6866 caucvgprprlemcl 6894 enreceq 6913 addsrpr 6922 mulsrpr 6923 0r 6927 1sr 6928 m1r 6929 addclsr 6930 mulclsr 6931 prsrcl 6960 addcnsr 7002 mulcnsr 7003 addcnsrec 7010 mulcnsrec 7011 pitonnlem2 7015 pitonn 7016 pitore 7018 recnnre 7019 axaddcl 7032 axmulcl 7034 xrlenlt 7177 cnrecnv 9797 eucalgf 10437 eucialg 10441 qredeu 10479 |
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