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Mirrors > Home > ILE Home > Th. List > relopabi | Unicode version |
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
relopabi.1 |
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Ref | Expression |
---|---|
relopabi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopabi.1 |
. . . 4
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2 | df-opab 3906 |
. . . 4
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3 | 1, 2 | eqtri 2109 |
. . 3
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4 | vex 2623 |
. . . . . . . 8
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5 | vex 2623 |
. . . . . . . 8
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6 | 4, 5 | opelvv 4501 |
. . . . . . 7
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7 | eleq1 2151 |
. . . . . . 7
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8 | 6, 7 | mpbiri 167 |
. . . . . 6
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9 | 8 | adantr 271 |
. . . . 5
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10 | 9 | exlimivv 1825 |
. . . 4
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11 | 10 | abssi 3097 |
. . 3
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12 | 3, 11 | eqsstri 3057 |
. 2
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13 | df-rel 4459 |
. 2
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14 | 12, 13 | mpbir 145 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-opab 3906 df-xp 4458 df-rel 4459 |
This theorem is referenced by: relopab 4577 reli 4578 rele 4579 relcnv 4823 cotr 4826 relco 4942 reloprab 5711 reldmoprab 5747 eqer 6338 ecopover 6404 ecopoverg 6407 relen 6515 reldom 6516 enq0er 7055 climrel 10729 brstruct 11564 |
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