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Mirrors > Home > ILE Home > Th. List > imainlem | Unicode version |
Description: One direction of imain 4981. This direction does not require . (Contributed by Jim Kingdon, 25-Dec-2018.) |
Ref | Expression |
---|---|
imainlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2312 | . . . . 5 | |
2 | elin 3126 | . . . . . . . . 9 | |
3 | 2 | anbi1i 431 | . . . . . . . 8 |
4 | anandir 525 | . . . . . . . 8 | |
5 | 3, 4 | bitri 173 | . . . . . . 7 |
6 | 5 | exbii 1496 | . . . . . 6 |
7 | 19.40 1522 | . . . . . 6 | |
8 | 6, 7 | sylbi 114 | . . . . 5 |
9 | 1, 8 | sylbi 114 | . . . 4 |
10 | df-rex 2312 | . . . . 5 | |
11 | df-rex 2312 | . . . . 5 | |
12 | 10, 11 | anbi12i 433 | . . . 4 |
13 | 9, 12 | sylibr 137 | . . 3 |
14 | 13 | ss2abi 3012 | . 2 |
15 | dfima2 4670 | . 2 | |
16 | dfima2 4670 | . . . 4 | |
17 | dfima2 4670 | . . . 4 | |
18 | 16, 17 | ineq12i 3136 | . . 3 |
19 | inab 3205 | . . 3 | |
20 | 18, 19 | eqtri 2060 | . 2 |
21 | 14, 15, 20 | 3sstr4i 2984 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wex 1381 wcel 1393 cab 2026 wrex 2307 cin 2916 wss 2917 class class class wbr 3764 cima 4348 |
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-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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 |
This theorem is referenced by: imain 4981 |
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