Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > onsucelsucexmid | Unicode version |
Description: The converse of onsucelsucr 4234 implies excluded middle. On the other hand, if is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4234 does hold, as seen at nnsucelsuc 6070. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
onsucelsucexmid.1 |
Ref | Expression |
---|---|
onsucelsucexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsucelsucexmidlem1 4253 | . . . 4 | |
2 | 0elon 4129 | . . . . . 6 | |
3 | onsucelsucexmidlem 4254 | . . . . . 6 | |
4 | 2, 3 | pm3.2i 257 | . . . . 5 |
5 | onsucelsucexmid.1 | . . . . 5 | |
6 | eleq1 2100 | . . . . . . 7 | |
7 | suceq 4139 | . . . . . . . 8 | |
8 | 7 | eleq1d 2106 | . . . . . . 7 |
9 | 6, 8 | imbi12d 223 | . . . . . 6 |
10 | eleq2 2101 | . . . . . . 7 | |
11 | suceq 4139 | . . . . . . . 8 | |
12 | 11 | eleq2d 2107 | . . . . . . 7 |
13 | 10, 12 | imbi12d 223 | . . . . . 6 |
14 | 9, 13 | rspc2va 2663 | . . . . 5 |
15 | 4, 5, 14 | mp2an 402 | . . . 4 |
16 | 1, 15 | ax-mp 7 | . . 3 |
17 | elsuci 4140 | . . 3 | |
18 | 16, 17 | ax-mp 7 | . 2 |
19 | suc0 4148 | . . . . . 6 | |
20 | p0ex 3939 | . . . . . . 7 | |
21 | 20 | prid2 3477 | . . . . . 6 |
22 | 19, 21 | eqeltri 2110 | . . . . 5 |
23 | eqeq1 2046 | . . . . . . 7 | |
24 | 23 | orbi1d 705 | . . . . . 6 |
25 | 24 | elrab3 2699 | . . . . 5 |
26 | 22, 25 | ax-mp 7 | . . . 4 |
27 | 0ex 3884 | . . . . . . 7 | |
28 | nsuceq0g 4155 | . . . . . . 7 | |
29 | 27, 28 | ax-mp 7 | . . . . . 6 |
30 | df-ne 2206 | . . . . . 6 | |
31 | 29, 30 | mpbi 133 | . . . . 5 |
32 | pm2.53 641 | . . . . 5 | |
33 | 31, 32 | mpi 15 | . . . 4 |
34 | 26, 33 | sylbi 114 | . . 3 |
35 | 19 | eqeq1i 2047 | . . . . 5 |
36 | 19 | eqeq1i 2047 | . . . . . . . 8 |
37 | 31, 36 | mtbi 595 | . . . . . . 7 |
38 | 20 | elsn 3391 | . . . . . . 7 |
39 | 37, 38 | mtbir 596 | . . . . . 6 |
40 | eleq2 2101 | . . . . . 6 | |
41 | 39, 40 | mtbii 599 | . . . . 5 |
42 | 35, 41 | sylbi 114 | . . . 4 |
43 | olc 632 | . . . . 5 | |
44 | eqeq1 2046 | . . . . . . . 8 | |
45 | 44 | orbi1d 705 | . . . . . . 7 |
46 | 45 | elrab3 2699 | . . . . . 6 |
47 | 21, 46 | ax-mp 7 | . . . . 5 |
48 | 43, 47 | sylibr 137 | . . . 4 |
49 | 42, 48 | nsyl 558 | . . 3 |
50 | 34, 49 | orim12i 676 | . 2 |
51 | 18, 50 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 wne 2204 wral 2306 crab 2310 cvv 2557 c0 3224 csn 3375 cpr 3376 con0 4100 csuc 4102 |
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-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 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 |
This theorem is referenced by: ordsucunielexmid 4256 |
Copyright terms: Public domain | W3C validator |