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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 14010 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 14005 | . . 3 Ind | |
2 | biimp 117 | . . . . . . 7 | |
3 | 2 | alimi 1448 | . . . . . 6 |
4 | df-ral 2453 | . . . . . 6 | |
5 | 3, 4 | sylibr 133 | . . . . 5 |
6 | bj-inf2vn.1 | . . . . . 6 BOUNDED | |
7 | bdcv 13883 | . . . . . 6 BOUNDED | |
8 | 6, 7 | bj-inf2vnlem3 14007 | . . . . 5 Ind |
9 | 5, 8 | syl 14 | . . . 4 Ind |
10 | 9 | alrimiv 1867 | . . 3 Ind |
11 | 1, 10 | jca 304 | . 2 Ind Ind |
12 | bj-om 13972 | . 2 Ind Ind | |
13 | 11, 12 | syl5ibr 155 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 wal 1346 wceq 1348 wcel 2141 wral 2448 wrex 2449 wss 3121 c0 3414 csuc 4350 com 4574 BOUNDED wbdc 13875 Ind wind 13961 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-nul 4115 ax-pr 4194 ax-un 4418 ax-bd0 13848 ax-bdim 13849 ax-bdor 13851 ax-bdex 13854 ax-bdeq 13855 ax-bdel 13856 ax-bdsb 13857 ax-bdsep 13919 ax-bdsetind 14003 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-iom 4575 df-bdc 13876 df-bj-ind 13962 |
This theorem is referenced by: bj-omex2 14012 bj-nn0sucALT 14013 |
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