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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2703 |
. 2
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2 | elisset 2703 |
. 2
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3 | ax-17 1507 |
. . . 4
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4 | 19.29r 1601 |
. . . 4
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5 | 3, 4 | sylan2 284 |
. . 3
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6 | ax-17 1507 |
. . . . 5
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7 | 19.29 1600 |
. . . . 5
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8 | 6, 7 | sylan 281 |
. . . 4
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9 | 8 | eximi 1580 |
. . 3
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10 | ineq12 3277 |
. . . . 5
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11 | 10 | 2eximi 1581 |
. . . 4
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12 | dfin5 3083 |
. . . . . . 7
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13 | vex 2692 |
. . . . . . . 8
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14 | ax-bdel 13190 |
. . . . . . . . 9
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15 | bdcv 13217 |
. . . . . . . . 9
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16 | 14, 15 | bdrabexg 13275 |
. . . . . . . 8
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17 | 13, 16 | ax-mp 5 |
. . . . . . 7
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18 | 12, 17 | eqeltri 2213 |
. . . . . 6
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19 | eleq1 2203 |
. . . . . 6
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20 | 18, 19 | mpbii 147 |
. . . . 5
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21 | 20 | exlimivv 1869 |
. . . 4
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22 | 11, 21 | syl 14 |
. . 3
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23 | 5, 9, 22 | 3syl 17 |
. 2
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24 | 1, 2, 23 | syl2an 287 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-bd0 13182 ax-bdan 13184 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-in 3082 df-ss 3089 df-bdc 13210 |
This theorem is referenced by: speano5 13313 |
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