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Mirrors > Home > ILE Home > Th. List > onsucsssucexmid | Unicode version |
Description: The converse of onsucsssucr 4235 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 3025 | . . . . . 6 | |
2 | ordtriexmidlem 4245 | . . . . . . 7 | |
3 | sseq1 2966 | . . . . . . . . 9 | |
4 | suceq 4139 | . . . . . . . . . 10 | |
5 | 4 | sseq1d 2972 | . . . . . . . . 9 |
6 | 3, 5 | imbi12d 223 | . . . . . . . 8 |
7 | suc0 4148 | . . . . . . . . . 10 | |
8 | 0elon 4129 | . . . . . . . . . . 11 | |
9 | 8 | onsuci 4242 | . . . . . . . . . 10 |
10 | 7, 9 | eqeltrri 2111 | . . . . . . . . 9 |
11 | p0ex 3939 | . . . . . . . . . 10 | |
12 | eleq1 2100 | . . . . . . . . . . . 12 | |
13 | 12 | anbi2d 437 | . . . . . . . . . . 11 |
14 | sseq2 2967 | . . . . . . . . . . . 12 | |
15 | suceq 4139 | . . . . . . . . . . . . 13 | |
16 | 15 | sseq2d 2973 | . . . . . . . . . . . 12 |
17 | 14, 16 | imbi12d 223 | . . . . . . . . . . 11 |
18 | 13, 17 | imbi12d 223 | . . . . . . . . . 10 |
19 | onsucsssucexmid.1 | . . . . . . . . . . 11 | |
20 | 19 | rspec2 2408 | . . . . . . . . . 10 |
21 | 11, 18, 20 | vtocl 2608 | . . . . . . . . 9 |
22 | 10, 21 | mpan2 401 | . . . . . . . 8 |
23 | 6, 22 | vtoclga 2619 | . . . . . . 7 |
24 | 2, 23 | ax-mp 7 | . . . . . 6 |
25 | 1, 24 | ax-mp 7 | . . . . 5 |
26 | 10 | onsuci 4242 | . . . . . . 7 |
27 | 26 | onordi 4163 | . . . . . 6 |
28 | ordelsuc 4231 | . . . . . 6 | |
29 | 2, 27, 28 | mp2an 402 | . . . . 5 |
30 | 25, 29 | mpbir 134 | . . . 4 |
31 | elsucg 4141 | . . . . 5 | |
32 | 2, 31 | ax-mp 7 | . . . 4 |
33 | 30, 32 | mpbi 133 | . . 3 |
34 | elsni 3393 | . . . . 5 | |
35 | ordtriexmidlem2 4246 | . . . . 5 | |
36 | 34, 35 | syl 14 | . . . 4 |
37 | 0ex 3884 | . . . . 5 | |
38 | biidd 161 | . . . . 5 | |
39 | 37, 38 | rabsnt 3445 | . . . 4 |
40 | 36, 39 | orim12i 676 | . . 3 |
41 | 33, 40 | ax-mp 7 | . 2 |
42 | orcom 647 | . 2 | |
43 | 41, 42 | mpbi 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 wral 2306 crab 2310 wss 2917 c0 3224 csn 3375 word 4099 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-13 1404 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 ax-pr 3944 ax-un 4170 |
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-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: (None) |
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