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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdop | Unicode version |
Description: The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
bdop | BOUNDED |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdvsn 13546 | . . . 4 BOUNDED | |
2 | bdcpr 13543 | . . . . . . 7 BOUNDED | |
3 | 2 | bdss 13536 | . . . . . 6 BOUNDED |
4 | ax-bdel 13493 | . . . . . . . 8 BOUNDED | |
5 | ax-bdel 13493 | . . . . . . . 8 BOUNDED | |
6 | 4, 5 | ax-bdan 13487 | . . . . . . 7 BOUNDED |
7 | vex 2715 | . . . . . . . . . . 11 | |
8 | 7 | prid1 3667 | . . . . . . . . . 10 |
9 | ssel 3122 | . . . . . . . . . 10 | |
10 | 8, 9 | mpi 15 | . . . . . . . . 9 |
11 | vex 2715 | . . . . . . . . . . 11 | |
12 | 11 | prid2 3668 | . . . . . . . . . 10 |
13 | ssel 3122 | . . . . . . . . . 10 | |
14 | 12, 13 | mpi 15 | . . . . . . . . 9 |
15 | 10, 14 | jca 304 | . . . . . . . 8 |
16 | prssi 3716 | . . . . . . . 8 | |
17 | 15, 16 | impbii 125 | . . . . . . 7 |
18 | 6, 17 | bd0r 13497 | . . . . . 6 BOUNDED |
19 | 3, 18 | ax-bdan 13487 | . . . . 5 BOUNDED |
20 | eqss 3143 | . . . . 5 | |
21 | 19, 20 | bd0r 13497 | . . . 4 BOUNDED |
22 | 1, 21 | ax-bdor 13488 | . . 3 BOUNDED |
23 | vex 2715 | . . . 4 | |
24 | 23, 7, 11 | elop 4194 | . . 3 |
25 | 22, 24 | bd0r 13497 | . 2 BOUNDED |
26 | 25 | bdelir 13519 | 1 BOUNDED |
Colors of variables: wff set class |
Syntax hints: wa 103 wo 698 wceq 1335 wcel 2128 wss 3102 csn 3561 cpr 3562 cop 3564 BOUNDED wbdc 13512 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-bd0 13485 ax-bdan 13487 ax-bdor 13488 ax-bdal 13490 ax-bdeq 13492 ax-bdel 13493 ax-bdsb 13494 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3567 df-pr 3568 df-op 3570 df-bdc 13513 |
This theorem is referenced by: (None) |
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