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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-om | Unicode version |
Description: A set is equal to if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-om | Ind Ind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omind 13132 | . . . 4 Ind | |
2 | bj-indeq 13127 | . . . 4 Ind Ind | |
3 | 1, 2 | mpbiri 167 | . . 3 Ind |
4 | vex 2689 | . . . . . 6 | |
5 | bj-omssind 13133 | . . . . . 6 Ind | |
6 | 4, 5 | ax-mp 5 | . . . . 5 Ind |
7 | sseq1 3120 | . . . . 5 | |
8 | 6, 7 | syl5ibr 155 | . . . 4 Ind |
9 | 8 | alrimiv 1846 | . . 3 Ind |
10 | 3, 9 | jca 304 | . 2 Ind Ind |
11 | bj-ssom 13134 | . . . . . . 7 Ind | |
12 | 11 | biimpi 119 | . . . . . 6 Ind |
13 | 12 | adantl 275 | . . . . 5 Ind Ind |
14 | 13 | a1i 9 | . . . 4 Ind Ind |
15 | bj-omssind 13133 | . . . . 5 Ind | |
16 | 15 | adantrd 277 | . . . 4 Ind Ind |
17 | 14, 16 | jcad 305 | . . 3 Ind Ind |
18 | eqss 3112 | . . 3 | |
19 | 17, 18 | syl6ibr 161 | . 2 Ind Ind |
20 | 10, 19 | impbid2 142 | 1 Ind Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 cvv 2686 wss 3071 com 4504 Ind wind 13124 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 ax-pr 4131 ax-un 4355 ax-bd0 13011 ax-bdor 13014 ax-bdex 13017 ax-bdeq 13018 ax-bdel 13019 ax-bdsb 13020 ax-bdsep 13082 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 df-bdc 13039 df-bj-ind 13125 |
This theorem is referenced by: bj-2inf 13136 bj-inf2vn 13172 bj-inf2vn2 13173 |
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