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Mirrors > Home > ILE Home > Th. List > bnd2 | Unicode version |
Description: A variant of the Boundedness Axiom bnd 3925 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 2312 | . . . 4 | |
2 | 1 | ralbii 2330 | . . 3 |
3 | bnd2.1 | . . . 4 | |
4 | raleq 2505 | . . . . 5 | |
5 | raleq 2505 | . . . . . 6 | |
6 | 5 | exbidv 1706 | . . . . 5 |
7 | 4, 6 | imbi12d 223 | . . . 4 |
8 | bnd 3925 | . . . 4 | |
9 | 3, 7, 8 | vtocl 2608 | . . 3 |
10 | 2, 9 | sylbi 114 | . 2 |
11 | vex 2560 | . . . . 5 | |
12 | 11 | inex1 3891 | . . . 4 |
13 | inss2 3158 | . . . . . . 7 | |
14 | sseq1 2966 | . . . . . . 7 | |
15 | 13, 14 | mpbiri 157 | . . . . . 6 |
16 | 15 | biantrurd 289 | . . . . 5 |
17 | rexeq 2506 | . . . . . . 7 | |
18 | elin 3126 | . . . . . . . . . 10 | |
19 | 18 | anbi1i 431 | . . . . . . . . 9 |
20 | anass 381 | . . . . . . . . 9 | |
21 | 19, 20 | bitri 173 | . . . . . . . 8 |
22 | 21 | rexbii2 2335 | . . . . . . 7 |
23 | 17, 22 | syl6bb 185 | . . . . . 6 |
24 | 23 | ralbidv 2326 | . . . . 5 |
25 | 16, 24 | bitr3d 179 | . . . 4 |
26 | 12, 25 | spcev 2647 | . . 3 |
27 | 26 | exlimiv 1489 | . 2 |
28 | 10, 27 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 wral 2306 wrex 2307 cvv 2557 cin 2916 wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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