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Mirrors > Home > ILE Home > Th. List > bnd2 | Unicode version |
Description: A variant of the Boundedness Axiom bnd 3946 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.) |
Ref | Expression |
---|---|
bnd2.1 |
Ref | Expression |
---|---|
bnd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2354 | . . . 4 | |
2 | 1 | ralbii 2372 | . . 3 |
3 | bnd2.1 | . . . 4 | |
4 | raleq 2549 | . . . . 5 | |
5 | raleq 2549 | . . . . . 6 | |
6 | 5 | exbidv 1746 | . . . . 5 |
7 | 4, 6 | imbi12d 232 | . . . 4 |
8 | bnd 3946 | . . . 4 | |
9 | 3, 7, 8 | vtocl 2653 | . . 3 |
10 | 2, 9 | sylbi 119 | . 2 |
11 | vex 2604 | . . . . 5 | |
12 | 11 | inex1 3912 | . . . 4 |
13 | inss2 3187 | . . . . . . 7 | |
14 | sseq1 3020 | . . . . . . 7 | |
15 | 13, 14 | mpbiri 166 | . . . . . 6 |
16 | 15 | biantrurd 299 | . . . . 5 |
17 | rexeq 2550 | . . . . . . 7 | |
18 | elin 3155 | . . . . . . . . . 10 | |
19 | 18 | anbi1i 445 | . . . . . . . . 9 |
20 | anass 393 | . . . . . . . . 9 | |
21 | 19, 20 | bitri 182 | . . . . . . . 8 |
22 | 21 | rexbii2 2377 | . . . . . . 7 |
23 | 17, 22 | syl6bb 194 | . . . . . 6 |
24 | 23 | ralbidv 2368 | . . . . 5 |
25 | 16, 24 | bitr3d 188 | . . . 4 |
26 | 12, 25 | spcev 2692 | . . 3 |
27 | 26 | exlimiv 1529 | . 2 |
28 | 10, 27 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 wral 2348 wrex 2349 cvv 2601 cin 2972 wss 2973 |
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-coll 3893 ax-sep 3896 |
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-ral 2353 df-rex 2354 df-v 2603 df-in 2979 df-ss 2986 |
This theorem is referenced by: (None) |
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