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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-inf2vn | Unicode version |
Description: A sufficient condition for to be a set. See bj-inf2vn2 10770 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-inf2vn.1 | BOUNDED |
Ref | Expression |
---|---|
bj-inf2vn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-inf2vnlem1 10765 | . . 3 Ind | |
2 | bi1 116 | . . . . . . 7 | |
3 | 2 | alimi 1384 | . . . . . 6 |
4 | df-ral 2353 | . . . . . 6 | |
5 | 3, 4 | sylibr 132 | . . . . 5 |
6 | bj-inf2vn.1 | . . . . . 6 BOUNDED | |
7 | bdcv 10639 | . . . . . 6 BOUNDED | |
8 | 6, 7 | bj-inf2vnlem3 10767 | . . . . 5 Ind |
9 | 5, 8 | syl 14 | . . . 4 Ind |
10 | 9 | alrimiv 1795 | . . 3 Ind |
11 | 1, 10 | jca 300 | . 2 Ind Ind |
12 | bj-om 10732 | . 2 Ind Ind | |
13 | 11, 12 | syl5ibr 154 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wal 1282 wceq 1284 wcel 1433 wral 2348 wrex 2349 wss 2973 c0 3251 csuc 4120 com 4331 BOUNDED wbdc 10631 Ind wind 10721 |
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-in1 576 ax-in2 577 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-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 ax-pr 3964 ax-un 4188 ax-bd0 10604 ax-bdim 10605 ax-bdor 10607 ax-bdex 10610 ax-bdeq 10611 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 ax-bdsetind 10763 |
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-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332 df-bdc 10632 df-bj-ind 10722 |
This theorem is referenced by: bj-omex2 10772 bj-nn0sucALT 10773 |
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