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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-inf2vnlem1 | Unicode version |
Description: Lemma for bj-inf2vn 10099. 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 121 | . . . . 5 | |
2 | ax-io 630 | . . . . . 6 | |
3 | 2 | biimpi 113 | . . . . 5 |
4 | simpl 102 | . . . . . 6 | |
5 | eleq1 2100 | . . . . . 6 | |
6 | 4, 5 | mpbidi 140 | . . . . 5 |
7 | 1, 3, 6 | 3syl 17 | . . . 4 |
8 | 7 | alimi 1344 | . . 3 |
9 | exim 1490 | . . 3 | |
10 | 0ex 3884 | . . . . . 6 | |
11 | 10 | isseti 2563 | . . . . 5 |
12 | pm2.27 35 | . . . . 5 | |
13 | 11, 12 | ax-mp 7 | . . . 4 |
14 | bj-ex 9902 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 8, 9, 15 | 3syl 17 | . 2 |
17 | 3 | simprd 107 | . . . . . 6 |
18 | 1, 17 | syl 14 | . . . . 5 |
19 | 18 | alimi 1344 | . . . 4 |
20 | eqid 2040 | . . . . 5 | |
21 | suceq 4139 | . . . . . . 7 | |
22 | 21 | eqeq2d 2051 | . . . . . 6 |
23 | 22 | rspcev 2656 | . . . . 5 |
24 | 20, 23 | mpan2 401 | . . . 4 |
25 | vex 2560 | . . . . . 6 | |
26 | 25 | bj-sucex 10043 | . . . . 5 |
27 | eqeq1 2046 | . . . . . . 7 | |
28 | 27 | rexbidv 2327 | . . . . . 6 |
29 | eleq1 2100 | . . . . . 6 | |
30 | 28, 29 | imbi12d 223 | . . . . 5 |
31 | 26, 30 | spcv 2646 | . . . 4 |
32 | 19, 24, 31 | syl2im 34 | . . 3 |
33 | 32 | ralrimiv 2391 | . 2 |
34 | df-bj-ind 10051 | . 2 Ind | |
35 | 16, 33, 34 | sylanbrc 394 | 1 Ind |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wal 1241 wceq 1243 wex 1381 wcel 1393 wral 2306 wrex 2307 c0 3224 csuc 4102 Ind wind 10050 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdor 9936 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-sn 3381 df-pr 3382 df-uni 3581 df-suc 4108 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bj-inf2vn 10099 bj-inf2vn2 10100 |
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