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Mirrors > Home > ILE Home > Th. List > acexmidlem2 | Unicode version |
Description: Lemma for acexmid 5511. This builds on acexmidlem1 5508 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 4904 sense because it uses ordered pairs as described in opthreg 4280 rather than df-op 3384). The set is also found in onsucelsucexmidlem 4254. (Contributed by Jim Kingdon, 5-Aug-2019.) |
Ref | Expression |
---|---|
acexmidlem.a | |
acexmidlem.b | |
acexmidlem.c |
Ref | Expression |
---|---|
acexmidlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2311 | . . . . 5 | |
2 | 19.23v 1763 | . . . . 5 | |
3 | 1, 2 | bitr2i 174 | . . . 4 |
4 | acexmidlem.c | . . . . . . . . 9 | |
5 | 4 | eleq2i 2104 | . . . . . . . 8 |
6 | vex 2560 | . . . . . . . . 9 | |
7 | 6 | elpr 3396 | . . . . . . . 8 |
8 | 5, 7 | bitri 173 | . . . . . . 7 |
9 | onsucelsucexmidlem1 4253 | . . . . . . . . . . 11 | |
10 | acexmidlem.a | . . . . . . . . . . 11 | |
11 | 9, 10 | eleqtrri 2113 | . . . . . . . . . 10 |
12 | elex2 2570 | . . . . . . . . . 10 | |
13 | 11, 12 | ax-mp 7 | . . . . . . . . 9 |
14 | eleq2 2101 | . . . . . . . . . 10 | |
15 | 14 | exbidv 1706 | . . . . . . . . 9 |
16 | 13, 15 | mpbiri 157 | . . . . . . . 8 |
17 | p0ex 3939 | . . . . . . . . . . . . 13 | |
18 | 17 | prid2 3477 | . . . . . . . . . . . 12 |
19 | eqid 2040 | . . . . . . . . . . . . 13 | |
20 | 19 | orci 650 | . . . . . . . . . . . 12 |
21 | eqeq1 2046 | . . . . . . . . . . . . . 14 | |
22 | 21 | orbi1d 705 | . . . . . . . . . . . . 13 |
23 | 22 | elrab 2698 | . . . . . . . . . . . 12 |
24 | 18, 20, 23 | mpbir2an 849 | . . . . . . . . . . 11 |
25 | acexmidlem.b | . . . . . . . . . . 11 | |
26 | 24, 25 | eleqtrri 2113 | . . . . . . . . . 10 |
27 | elex2 2570 | . . . . . . . . . 10 | |
28 | 26, 27 | ax-mp 7 | . . . . . . . . 9 |
29 | eleq2 2101 | . . . . . . . . . 10 | |
30 | 29 | exbidv 1706 | . . . . . . . . 9 |
31 | 28, 30 | mpbiri 157 | . . . . . . . 8 |
32 | 16, 31 | jaoi 636 | . . . . . . 7 |
33 | 8, 32 | sylbi 114 | . . . . . 6 |
34 | pm2.27 35 | . . . . . 6 | |
35 | 33, 34 | syl 14 | . . . . 5 |
36 | 35 | imp 115 | . . . 4 |
37 | 3, 36 | sylan2br 272 | . . 3 |
38 | 37 | ralimiaa 2383 | . 2 |
39 | 10, 25, 4 | acexmidlem1 5508 | . 2 |
40 | 38, 39 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wex 1381 wcel 1393 wral 2306 wrex 2307 wreu 2308 crab 2310 c0 3224 csn 3375 cpr 3376 |
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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 df-iota 4867 df-riota 5468 |
This theorem is referenced by: acexmidlemv 5510 |
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