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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 13551 | . . . 4 Ind | |
2 | bj-indeq 13546 | . . . 4 Ind Ind | |
3 | 1, 2 | mpbiri 167 | . . 3 Ind |
4 | vex 2715 | . . . . . 6 | |
5 | bj-omssind 13552 | . . . . . 6 Ind | |
6 | 4, 5 | ax-mp 5 | . . . . 5 Ind |
7 | sseq1 3151 | . . . . 5 | |
8 | 6, 7 | syl5ibr 155 | . . . 4 Ind |
9 | 8 | alrimiv 1854 | . . 3 Ind |
10 | 3, 9 | jca 304 | . 2 Ind Ind |
11 | bj-ssom 13553 | . . . . . . 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 13552 | . . . . 5 Ind | |
16 | 15 | adantrd 277 | . . . 4 Ind Ind |
17 | 14, 16 | jcad 305 | . . 3 Ind Ind |
18 | eqss 3143 | . . 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 1333 wceq 1335 wcel 2128 cvv 2712 wss 3102 com 4550 Ind wind 13543 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-nul 4091 ax-pr 4170 ax-un 4394 ax-bd0 13430 ax-bdor 13433 ax-bdex 13436 ax-bdeq 13437 ax-bdel 13438 ax-bdsb 13439 ax-bdsep 13501 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 df-pr 3567 df-uni 3774 df-int 3809 df-suc 4332 df-iom 4551 df-bdc 13458 df-bj-ind 13544 |
This theorem is referenced by: bj-2inf 13555 bj-inf2vn 13591 bj-inf2vn2 13592 |
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