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Mirrors > Home > ILE Home > Th. List > opelxp | Unicode version |
Description: Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 4557 | . 2 | |
2 | vex 2689 | . . . . . . 7 | |
3 | vex 2689 | . . . . . . 7 | |
4 | 2, 3 | opth2 4162 | . . . . . 6 |
5 | eleq1 2202 | . . . . . . 7 | |
6 | eleq1 2202 | . . . . . . 7 | |
7 | 5, 6 | bi2anan9 595 | . . . . . 6 |
8 | 4, 7 | sylbi 120 | . . . . 5 |
9 | 8 | biimprcd 159 | . . . 4 |
10 | 9 | rexlimivv 2555 | . . 3 |
11 | eqid 2139 | . . . 4 | |
12 | opeq1 3705 | . . . . . 6 | |
13 | 12 | eqeq2d 2151 | . . . . 5 |
14 | opeq2 3706 | . . . . . 6 | |
15 | 14 | eqeq2d 2151 | . . . . 5 |
16 | 13, 15 | rspc2ev 2804 | . . . 4 |
17 | 11, 16 | mp3an3 1304 | . . 3 |
18 | 10, 17 | impbii 125 | . 2 |
19 | 1, 18 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 cop 3530 cxp 4537 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 |
This theorem is referenced by: brxp 4570 opelxpi 4571 opelxp1 4573 opelxp2 4574 opthprc 4590 elxp3 4593 opeliunxp 4594 optocl 4615 xpiindim 4676 opelres 4824 resiexg 4864 codir 4927 qfto 4928 xpmlem 4959 rnxpid 4973 ssrnres 4981 dfco2 5038 relssdmrn 5059 ressn 5079 opelf 5294 fnovex 5804 oprab4 5842 resoprab 5867 elmpocl 5968 fo1stresm 6059 fo2ndresm 6060 dfoprab4 6090 xporderlem 6128 f1od2 6132 brecop 6519 xpdom2 6725 djulclb 6940 djuss 6955 enq0enq 7246 enq0sym 7247 enq0tr 7249 nqnq0pi 7253 nnnq0lem1 7261 elinp 7289 genipv 7324 prsrlem1 7557 gt0srpr 7563 opelcn 7641 opelreal 7642 elreal2 7645 frecuzrdgrrn 10188 frec2uzrdg 10189 frecuzrdgrcl 10190 frecuzrdgsuc 10194 frecuzrdgrclt 10195 frecuzrdgsuctlem 10203 fisumcom2 11214 sqpweven 11860 2sqpwodd 11861 phimullem 11908 txuni2 12435 txcnp 12450 txcnmpt 12452 txdis1cn 12457 txlm 12458 xmeterval 12614 limccnp2lem 12824 limccnp2cntop 12825 |
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