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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-inex | Unicode version |
Description: The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-inex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2613 | . 2 | |
2 | elisset 2613 | . 2 | |
3 | ax-17 1459 | . . . 4 | |
4 | 19.29r 1552 | . . . 4 | |
5 | 3, 4 | sylan2 280 | . . 3 |
6 | ax-17 1459 | . . . . 5 | |
7 | 19.29 1551 | . . . . 5 | |
8 | 6, 7 | sylan 277 | . . . 4 |
9 | 8 | eximi 1531 | . . 3 |
10 | ineq12 3162 | . . . . 5 | |
11 | 10 | 2eximi 1532 | . . . 4 |
12 | dfin5 2980 | . . . . . . 7 | |
13 | vex 2604 | . . . . . . . 8 | |
14 | ax-bdel 10612 | . . . . . . . . 9 BOUNDED | |
15 | bdcv 10639 | . . . . . . . . 9 BOUNDED | |
16 | 14, 15 | bdrabexg 10697 | . . . . . . . 8 |
17 | 13, 16 | ax-mp 7 | . . . . . . 7 |
18 | 12, 17 | eqeltri 2151 | . . . . . 6 |
19 | eleq1 2141 | . . . . . 6 | |
20 | 18, 19 | mpbii 146 | . . . . 5 |
21 | 20 | exlimivv 1817 | . . . 4 |
22 | 11, 21 | syl 14 | . . 3 |
23 | 5, 9, 22 | 3syl 17 | . 2 |
24 | 1, 2, 23 | syl2an 283 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wex 1421 wcel 1433 crab 2352 cvv 2601 cin 2972 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-bd0 10604 ax-bdan 10606 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 df-v 2603 df-in 2979 df-ss 2986 df-bdc 10632 |
This theorem is referenced by: speano5 10739 |
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