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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 2695 | . 2 | |
2 | elisset 2695 | . 2 | |
3 | ax-17 1506 | . . . 4 | |
4 | 19.29r 1600 | . . . 4 | |
5 | 3, 4 | sylan2 284 | . . 3 |
6 | ax-17 1506 | . . . . 5 | |
7 | 19.29 1599 | . . . . 5 | |
8 | 6, 7 | sylan 281 | . . . 4 |
9 | 8 | eximi 1579 | . . 3 |
10 | ineq12 3267 | . . . . 5 | |
11 | 10 | 2eximi 1580 | . . . 4 |
12 | dfin5 3073 | . . . . . . 7 | |
13 | vex 2684 | . . . . . . . 8 | |
14 | ax-bdel 13008 | . . . . . . . . 9 BOUNDED | |
15 | bdcv 13035 | . . . . . . . . 9 BOUNDED | |
16 | 14, 15 | bdrabexg 13093 | . . . . . . . 8 |
17 | 13, 16 | ax-mp 5 | . . . . . . 7 |
18 | 12, 17 | eqeltri 2210 | . . . . . 6 |
19 | eleq1 2200 | . . . . . 6 | |
20 | 18, 19 | mpbii 147 | . . . . 5 |
21 | 20 | exlimivv 1868 | . . . 4 |
22 | 11, 21 | syl 14 | . . 3 |
23 | 5, 9, 22 | 3syl 17 | . 2 |
24 | 1, 2, 23 | syl2an 287 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wex 1468 wcel 1480 crab 2418 cvv 2681 cin 3065 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-bd0 13000 ax-bdan 13002 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-in 3072 df-ss 3079 df-bdc 13028 |
This theorem is referenced by: speano5 13131 |
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