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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-inf2vnlem1 | Unicode version |
Description: Lemma for bj-inf2vn 13172. 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 | bi2 129 | . . . . 5 | |
2 | jaob 699 | . . . . . 6 | |
3 | 2 | biimpi 119 | . . . . 5 |
4 | simpl 108 | . . . . . 6 | |
5 | eleq1 2202 | . . . . . 6 | |
6 | 4, 5 | mpbidi 150 | . . . . 5 |
7 | 1, 3, 6 | 3syl 17 | . . . 4 |
8 | 7 | alimi 1431 | . . 3 |
9 | exim 1578 | . . 3 | |
10 | 0ex 4055 | . . . . . 6 | |
11 | 10 | isseti 2694 | . . . . 5 |
12 | pm2.27 40 | . . . . 5 | |
13 | 11, 12 | ax-mp 5 | . . . 4 |
14 | bj-ex 12969 | . . . 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 1431 | . . . 4 |
20 | eqid 2139 | . . . . 5 | |
21 | suceq 4324 | . . . . . . 7 | |
22 | 21 | eqeq2d 2151 | . . . . . 6 |
23 | 22 | rspcev 2789 | . . . . 5 |
24 | 20, 23 | mpan2 421 | . . . 4 |
25 | vex 2689 | . . . . . 6 | |
26 | 25 | bj-sucex 13121 | . . . . 5 |
27 | eqeq1 2146 | . . . . . . 7 | |
28 | 27 | rexbidv 2438 | . . . . . 6 |
29 | eleq1 2202 | . . . . . 6 | |
30 | 28, 29 | imbi12d 233 | . . . . 5 |
31 | 26, 30 | spcv 2779 | . . . 4 |
32 | 19, 24, 31 | syl2im 38 | . . 3 |
33 | 32 | ralrimiv 2504 | . 2 |
34 | df-bj-ind 13125 | . 2 Ind | |
35 | 16, 33, 34 | sylanbrc 413 | 1 Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wal 1329 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 c0 3363 csuc 4287 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-v 2688 df-dif 3073 df-un 3075 df-nul 3364 df-sn 3533 df-pr 3534 df-uni 3737 df-suc 4293 df-bdc 13039 df-bj-ind 13125 |
This theorem is referenced by: bj-inf2vn 13172 bj-inf2vn2 13173 |
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