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Mirrors > Home > ILE Home > Th. List > acexmidlem2 | Unicode version |
Description: Lemma for acexmid 5826. This builds on acexmidlem1 5823 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 5175 sense because it uses ordered pairs as described in opthreg 4518 rather than df-op 3570). The set is also found in onsucelsucexmidlem 4491. (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 2440 | . . . . 5 | |
2 | 19.23v 1863 | . . . . 5 | |
3 | 1, 2 | bitr2i 184 | . . . 4 |
4 | acexmidlem.c | . . . . . . . . 9 | |
5 | 4 | eleq2i 2224 | . . . . . . . 8 |
6 | vex 2715 | . . . . . . . . 9 | |
7 | 6 | elpr 3582 | . . . . . . . 8 |
8 | 5, 7 | bitri 183 | . . . . . . 7 |
9 | onsucelsucexmidlem1 4490 | . . . . . . . . . . 11 | |
10 | acexmidlem.a | . . . . . . . . . . 11 | |
11 | 9, 10 | eleqtrri 2233 | . . . . . . . . . 10 |
12 | elex2 2728 | . . . . . . . . . 10 | |
13 | 11, 12 | ax-mp 5 | . . . . . . . . 9 |
14 | eleq2 2221 | . . . . . . . . . 10 | |
15 | 14 | exbidv 1805 | . . . . . . . . 9 |
16 | 13, 15 | mpbiri 167 | . . . . . . . 8 |
17 | p0ex 4152 | . . . . . . . . . . . . 13 | |
18 | 17 | prid2 3668 | . . . . . . . . . . . 12 |
19 | eqid 2157 | . . . . . . . . . . . . 13 | |
20 | 19 | orci 721 | . . . . . . . . . . . 12 |
21 | eqeq1 2164 | . . . . . . . . . . . . . 14 | |
22 | 21 | orbi1d 781 | . . . . . . . . . . . . 13 |
23 | 22 | elrab 2868 | . . . . . . . . . . . 12 |
24 | 18, 20, 23 | mpbir2an 927 | . . . . . . . . . . 11 |
25 | acexmidlem.b | . . . . . . . . . . 11 | |
26 | 24, 25 | eleqtrri 2233 | . . . . . . . . . 10 |
27 | elex2 2728 | . . . . . . . . . 10 | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 |
29 | eleq2 2221 | . . . . . . . . . 10 | |
30 | 29 | exbidv 1805 | . . . . . . . . 9 |
31 | 28, 30 | mpbiri 167 | . . . . . . . 8 |
32 | 16, 31 | jaoi 706 | . . . . . . 7 |
33 | 8, 32 | sylbi 120 | . . . . . 6 |
34 | pm2.27 40 | . . . . . 6 | |
35 | 33, 34 | syl 14 | . . . . 5 |
36 | 35 | imp 123 | . . . 4 |
37 | 3, 36 | sylan2br 286 | . . 3 |
38 | 37 | ralimiaa 2519 | . 2 |
39 | 10, 25, 4 | acexmidlem1 5823 | . 2 |
40 | 38, 39 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wal 1333 wceq 1335 wex 1472 wcel 2128 wral 2435 wrex 2436 wreu 2437 crab 2439 c0 3395 csn 3561 cpr 3562 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-nul 4093 ax-pow 4138 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-uni 3775 df-tr 4066 df-iord 4329 df-on 4331 df-suc 4334 df-iota 5138 df-riota 5783 |
This theorem is referenced by: acexmidlemv 5825 |
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