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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3900 | . . . . . 6 | |
2 | elopab 4150 | . . . . . 6 | |
3 | 1, 2 | bitri 183 | . . . . 5 |
4 | exsimpl 1581 | . . . . . 6 | |
5 | 4 | eximi 1564 | . . . . 5 |
6 | 3, 5 | sylbi 120 | . . . 4 |
7 | vex 2663 | . . . . . . . 8 | |
8 | vex 2663 | . . . . . . . 8 | |
9 | 7, 8 | opth 4129 | . . . . . . 7 |
10 | 9 | biimpi 119 | . . . . . 6 |
11 | 10 | eqcoms 2120 | . . . . 5 |
12 | 11 | 2eximi 1565 | . . . 4 |
13 | 6, 12 | syl 14 | . . 3 |
14 | eeanv 1884 | . . 3 | |
15 | 13, 14 | sylib 121 | . 2 |
16 | isset 2666 | . . 3 | |
17 | isset 2666 | . . 3 | |
18 | 16, 17 | anbi12i 455 | . 2 |
19 | 15, 18 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 cvv 2660 cop 3500 class class class wbr 3899 copab 3958 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 |
This theorem is referenced by: (None) |
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