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Mirrors > Home > ILE Home > Th. List > brabvv | Unicode version |
Description: If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.) |
Ref | Expression |
---|---|
brabvv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3896 |
. . . . . 6
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2 | elopab 4140 |
. . . . . 6
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3 | 1, 2 | bitri 183 |
. . . . 5
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4 | exsimpl 1579 |
. . . . . 6
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5 | 4 | eximi 1562 |
. . . . 5
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6 | 3, 5 | sylbi 120 |
. . . 4
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7 | vex 2660 |
. . . . . . . 8
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8 | vex 2660 |
. . . . . . . 8
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9 | 7, 8 | opth 4119 |
. . . . . . 7
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10 | 9 | biimpi 119 |
. . . . . 6
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11 | 10 | eqcoms 2118 |
. . . . 5
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12 | 11 | 2eximi 1563 |
. . . 4
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13 | 6, 12 | syl 14 |
. . 3
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14 | eeanv 1882 |
. . 3
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15 | 13, 14 | sylib 121 |
. 2
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16 | isset 2663 |
. . 3
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17 | isset 2663 |
. . 3
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18 | 16, 17 | anbi12i 453 |
. 2
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19 | 15, 18 | sylibr 133 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 |
This theorem is referenced by: (None) |
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