Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > acexmidlem2 | Unicode version |
Description: Lemma for acexmid 5852. This builds on acexmidlem1 5849 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 5200 sense because it uses ordered pairs as described in opthreg 4540 rather than df-op 3592). The set is also found in onsucelsucexmidlem 4513. (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 2453 | . . . . 5 | |
2 | 19.23v 1876 | . . . . 5 | |
3 | 1, 2 | bitr2i 184 | . . . 4 |
4 | acexmidlem.c | . . . . . . . . 9 | |
5 | 4 | eleq2i 2237 | . . . . . . . 8 |
6 | vex 2733 | . . . . . . . . 9 | |
7 | 6 | elpr 3604 | . . . . . . . 8 |
8 | 5, 7 | bitri 183 | . . . . . . 7 |
9 | onsucelsucexmidlem1 4512 | . . . . . . . . . . 11 | |
10 | acexmidlem.a | . . . . . . . . . . 11 | |
11 | 9, 10 | eleqtrri 2246 | . . . . . . . . . 10 |
12 | elex2 2746 | . . . . . . . . . 10 | |
13 | 11, 12 | ax-mp 5 | . . . . . . . . 9 |
14 | eleq2 2234 | . . . . . . . . . 10 | |
15 | 14 | exbidv 1818 | . . . . . . . . 9 |
16 | 13, 15 | mpbiri 167 | . . . . . . . 8 |
17 | p0ex 4174 | . . . . . . . . . . . . 13 | |
18 | 17 | prid2 3690 | . . . . . . . . . . . 12 |
19 | eqid 2170 | . . . . . . . . . . . . 13 | |
20 | 19 | orci 726 | . . . . . . . . . . . 12 |
21 | eqeq1 2177 | . . . . . . . . . . . . . 14 | |
22 | 21 | orbi1d 786 | . . . . . . . . . . . . 13 |
23 | 22 | elrab 2886 | . . . . . . . . . . . 12 |
24 | 18, 20, 23 | mpbir2an 937 | . . . . . . . . . . 11 |
25 | acexmidlem.b | . . . . . . . . . . 11 | |
26 | 24, 25 | eleqtrri 2246 | . . . . . . . . . 10 |
27 | elex2 2746 | . . . . . . . . . 10 | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 |
29 | eleq2 2234 | . . . . . . . . . 10 | |
30 | 29 | exbidv 1818 | . . . . . . . . 9 |
31 | 28, 30 | mpbiri 167 | . . . . . . . 8 |
32 | 16, 31 | jaoi 711 | . . . . . . 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 2532 | . 2 |
39 | 10, 25, 4 | acexmidlem1 5849 | . 2 |
40 | 38, 39 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wal 1346 wceq 1348 wex 1485 wcel 2141 wral 2448 wrex 2449 wreu 2450 crab 2452 c0 3414 csn 3583 cpr 3584 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-iota 5160 df-riota 5809 |
This theorem is referenced by: acexmidlemv 5851 |
Copyright terms: Public domain | W3C validator |