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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 4019 |
. . . . . 6
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2 | elopab 4276 |
. . . . . 6
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3 | 1, 2 | bitri 184 |
. . . . 5
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4 | exsimpl 1628 |
. . . . . 6
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5 | 4 | eximi 1611 |
. . . . 5
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6 | 3, 5 | sylbi 121 |
. . . 4
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7 | vex 2755 |
. . . . . . . 8
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8 | vex 2755 |
. . . . . . . 8
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9 | 7, 8 | opth 4255 |
. . . . . . 7
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10 | 9 | biimpi 120 |
. . . . . 6
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11 | 10 | eqcoms 2192 |
. . . . 5
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12 | 11 | 2eximi 1612 |
. . . 4
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13 | 6, 12 | syl 14 |
. . 3
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14 | eeanv 1944 |
. . 3
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15 | 13, 14 | sylib 122 |
. 2
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16 | isset 2758 |
. . 3
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17 | isset 2758 |
. . 3
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18 | 16, 17 | anbi12i 460 |
. 2
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19 | 15, 18 | sylibr 134 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 |
This theorem is referenced by: (None) |
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