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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-inf2vnlem1 | Unicode version |
Description: Lemma for bj-inf2vn 14009. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-inf2vnlem1 | Ind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpr 129 | . . . . 5 | |
2 | jaob 705 | . . . . . 6 | |
3 | 2 | biimpi 119 | . . . . 5 |
4 | simpl 108 | . . . . . 6 | |
5 | eleq1 2233 | . . . . . 6 | |
6 | 4, 5 | mpbidi 150 | . . . . 5 |
7 | 1, 3, 6 | 3syl 17 | . . . 4 |
8 | 7 | alimi 1448 | . . 3 |
9 | exim 1592 | . . 3 | |
10 | 0ex 4116 | . . . . . 6 | |
11 | 10 | isseti 2738 | . . . . 5 |
12 | pm2.27 40 | . . . . 5 | |
13 | 11, 12 | ax-mp 5 | . . . 4 |
14 | bj-ex 13797 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 8, 9, 15 | 3syl 17 | . 2 |
17 | 3 | simprd 113 | . . . . . 6 |
18 | 1, 17 | syl 14 | . . . . 5 |
19 | 18 | alimi 1448 | . . . 4 |
20 | eqid 2170 | . . . . 5 | |
21 | suceq 4387 | . . . . . . 7 | |
22 | 21 | eqeq2d 2182 | . . . . . 6 |
23 | 22 | rspcev 2834 | . . . . 5 |
24 | 20, 23 | mpan2 423 | . . . 4 |
25 | vex 2733 | . . . . . 6 | |
26 | 25 | bj-sucex 13958 | . . . . 5 |
27 | eqeq1 2177 | . . . . . . 7 | |
28 | 27 | rexbidv 2471 | . . . . . 6 |
29 | eleq1 2233 | . . . . . 6 | |
30 | 28, 29 | imbi12d 233 | . . . . 5 |
31 | 26, 30 | spcv 2824 | . . . 4 |
32 | 19, 24, 31 | syl2im 38 | . . 3 |
33 | 32 | ralrimiv 2542 | . 2 |
34 | df-bj-ind 13962 | . 2 Ind | |
35 | 16, 33, 34 | sylanbrc 415 | 1 Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 wal 1346 wceq 1348 wex 1485 wcel 2141 wral 2448 wrex 2449 c0 3414 csuc 4350 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-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-sn 3589 df-pr 3590 df-uni 3797 df-suc 4356 df-bdc 13876 df-bj-ind 13962 |
This theorem is referenced by: bj-inf2vn 14009 bj-inf2vn2 14010 |
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