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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 4306 |
. 2
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2 | vex 2554 |
. . . . . . 7
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3 | vex 2554 |
. . . . . . 7
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4 | 2, 3 | opth2 3968 |
. . . . . 6
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5 | eleq1 2097 |
. . . . . . 7
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6 | eleq1 2097 |
. . . . . . 7
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7 | 5, 6 | bi2anan9 538 |
. . . . . 6
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8 | 4, 7 | sylbi 114 |
. . . . 5
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9 | 8 | biimprcd 149 |
. . . 4
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10 | 9 | rexlimivv 2432 |
. . 3
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11 | eqid 2037 |
. . . 4
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12 | opeq1 3540 |
. . . . . 6
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13 | 12 | eqeq2d 2048 |
. . . . 5
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14 | opeq2 3541 |
. . . . . 6
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15 | 14 | eqeq2d 2048 |
. . . . 5
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16 | 13, 15 | rspc2ev 2658 |
. . . 4
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17 | 11, 16 | mp3an3 1220 |
. . 3
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18 | 10, 17 | impbii 117 |
. 2
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19 | 1, 18 | bitri 173 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 |
This theorem is referenced by: brxp 4318 opelxpi 4319 opelxp1 4320 opelxp2 4321 opthprc 4334 elxp3 4337 opeliunxp 4338 optocl 4359 xpiindim 4416 opelres 4560 resiexg 4596 codir 4656 qfto 4657 xpmlem 4687 rnxpid 4698 ssrnres 4706 dfco2 4763 relssdmrn 4784 ressn 4801 opelf 5005 fnovex 5481 oprab4 5517 resoprab 5539 elmpt2cl 5640 fo1stresm 5730 fo2ndresm 5731 dfoprab4 5760 xporderlem 5793 brecop 6132 xpdom2 6241 enq0enq 6414 enq0sym 6415 enq0tr 6417 nqnq0pi 6421 nnnq0lem1 6429 elinp 6457 genipv 6492 prsrlem1 6670 gt0srpr 6676 opelcn 6725 opelreal 6726 elreal2 6728 frecuzrdgrrn 8875 frec2uzrdg 8876 frecuzrdgrom 8877 frecuzrdgsuc 8882 |
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