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Mirrors > Home > ILE Home > Th. List > onsucsssucexmid | Unicode version |
Description: The converse of onsucsssucr 4253 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Ref | Expression |
---|---|
onsucsssucexmid.1 |
Ref | Expression |
---|---|
onsucsssucexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3079 | . . . . . 6 | |
2 | ordtriexmidlem 4263 | . . . . . . 7 | |
3 | sseq1 3020 | . . . . . . . . 9 | |
4 | suceq 4157 | . . . . . . . . . 10 | |
5 | 4 | sseq1d 3026 | . . . . . . . . 9 |
6 | 3, 5 | imbi12d 232 | . . . . . . . 8 |
7 | suc0 4166 | . . . . . . . . . 10 | |
8 | 0elon 4147 | . . . . . . . . . . 11 | |
9 | 8 | onsuci 4260 | . . . . . . . . . 10 |
10 | 7, 9 | eqeltrri 2152 | . . . . . . . . 9 |
11 | p0ex 3959 | . . . . . . . . . 10 | |
12 | eleq1 2141 | . . . . . . . . . . . 12 | |
13 | 12 | anbi2d 451 | . . . . . . . . . . 11 |
14 | sseq2 3021 | . . . . . . . . . . . 12 | |
15 | suceq 4157 | . . . . . . . . . . . . 13 | |
16 | 15 | sseq2d 3027 | . . . . . . . . . . . 12 |
17 | 14, 16 | imbi12d 232 | . . . . . . . . . . 11 |
18 | 13, 17 | imbi12d 232 | . . . . . . . . . 10 |
19 | onsucsssucexmid.1 | . . . . . . . . . . 11 | |
20 | 19 | rspec2 2450 | . . . . . . . . . 10 |
21 | 11, 18, 20 | vtocl 2653 | . . . . . . . . 9 |
22 | 10, 21 | mpan2 415 | . . . . . . . 8 |
23 | 6, 22 | vtoclga 2664 | . . . . . . 7 |
24 | 2, 23 | ax-mp 7 | . . . . . 6 |
25 | 1, 24 | ax-mp 7 | . . . . 5 |
26 | 10 | onsuci 4260 | . . . . . . 7 |
27 | 26 | onordi 4181 | . . . . . 6 |
28 | ordelsuc 4249 | . . . . . 6 | |
29 | 2, 27, 28 | mp2an 416 | . . . . 5 |
30 | 25, 29 | mpbir 144 | . . . 4 |
31 | elsucg 4159 | . . . . 5 | |
32 | 2, 31 | ax-mp 7 | . . . 4 |
33 | 30, 32 | mpbi 143 | . . 3 |
34 | elsni 3416 | . . . . 5 | |
35 | ordtriexmidlem2 4264 | . . . . 5 | |
36 | 34, 35 | syl 14 | . . . 4 |
37 | 0ex 3905 | . . . . 5 | |
38 | biidd 170 | . . . . 5 | |
39 | 37, 38 | rabsnt 3467 | . . . 4 |
40 | 36, 39 | orim12i 708 | . . 3 |
41 | 33, 40 | ax-mp 7 | . 2 |
42 | orcom 679 | . 2 | |
43 | 41, 42 | mpbi 143 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 wral 2348 crab 2352 wss 2973 c0 3251 csn 3398 word 4117 con0 4118 csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: (None) |
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