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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 13969 | . . . 4 Ind | |
2 | bj-indeq 13964 | . . . 4 Ind Ind | |
3 | 1, 2 | mpbiri 167 | . . 3 Ind |
4 | vex 2733 | . . . . . 6 | |
5 | bj-omssind 13970 | . . . . . 6 Ind | |
6 | 4, 5 | ax-mp 5 | . . . . 5 Ind |
7 | sseq1 3170 | . . . . 5 | |
8 | 6, 7 | syl5ibr 155 | . . . 4 Ind |
9 | 8 | alrimiv 1867 | . . 3 Ind |
10 | 3, 9 | jca 304 | . 2 Ind Ind |
11 | bj-ssom 13971 | . . . . . . 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 13970 | . . . . 5 Ind | |
16 | 15 | adantrd 277 | . . . 4 Ind Ind |
17 | 14, 16 | jcad 305 | . . 3 Ind Ind |
18 | eqss 3162 | . . 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 1346 wceq 1348 wcel 2141 cvv 2730 wss 3121 com 4574 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-bdor 13851 ax-bdex 13854 ax-bdeq 13855 ax-bdel 13856 ax-bdsb 13857 ax-bdsep 13919 |
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-2inf 13973 bj-inf2vn 14009 bj-inf2vn2 14010 |
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