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Mirrors > Home > ILE Home > Th. List > onsucsssucexmid | Unicode version |
Description: The converse of onsucsssucr 4395 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 3152 | . . . . . 6 | |
2 | ordtriexmidlem 4405 | . . . . . . 7 | |
3 | sseq1 3090 | . . . . . . . . 9 | |
4 | suceq 4294 | . . . . . . . . . 10 | |
5 | 4 | sseq1d 3096 | . . . . . . . . 9 |
6 | 3, 5 | imbi12d 233 | . . . . . . . 8 |
7 | suc0 4303 | . . . . . . . . . 10 | |
8 | 0elon 4284 | . . . . . . . . . . 11 | |
9 | 8 | onsuci 4402 | . . . . . . . . . 10 |
10 | 7, 9 | eqeltrri 2191 | . . . . . . . . 9 |
11 | p0ex 4082 | . . . . . . . . . 10 | |
12 | eleq1 2180 | . . . . . . . . . . . 12 | |
13 | 12 | anbi2d 459 | . . . . . . . . . . 11 |
14 | sseq2 3091 | . . . . . . . . . . . 12 | |
15 | suceq 4294 | . . . . . . . . . . . . 13 | |
16 | 15 | sseq2d 3097 | . . . . . . . . . . . 12 |
17 | 14, 16 | imbi12d 233 | . . . . . . . . . . 11 |
18 | 13, 17 | imbi12d 233 | . . . . . . . . . 10 |
19 | onsucsssucexmid.1 | . . . . . . . . . . 11 | |
20 | 19 | rspec2 2498 | . . . . . . . . . 10 |
21 | 11, 18, 20 | vtocl 2714 | . . . . . . . . 9 |
22 | 10, 21 | mpan2 421 | . . . . . . . 8 |
23 | 6, 22 | vtoclga 2726 | . . . . . . 7 |
24 | 2, 23 | ax-mp 5 | . . . . . 6 |
25 | 1, 24 | ax-mp 5 | . . . . 5 |
26 | 10 | onsuci 4402 | . . . . . . 7 |
27 | 26 | onordi 4318 | . . . . . 6 |
28 | ordelsuc 4391 | . . . . . 6 | |
29 | 2, 27, 28 | mp2an 422 | . . . . 5 |
30 | 25, 29 | mpbir 145 | . . . 4 |
31 | elsucg 4296 | . . . . 5 | |
32 | 2, 31 | ax-mp 5 | . . . 4 |
33 | 30, 32 | mpbi 144 | . . 3 |
34 | elsni 3515 | . . . . 5 | |
35 | ordtriexmidlem2 4406 | . . . . 5 | |
36 | 34, 35 | syl 14 | . . . 4 |
37 | 0ex 4025 | . . . . 5 | |
38 | biidd 171 | . . . . 5 | |
39 | 37, 38 | rabsnt 3568 | . . . 4 |
40 | 36, 39 | orim12i 733 | . . 3 |
41 | 33, 40 | ax-mp 5 | . 2 |
42 | orcom 702 | . 2 | |
43 | 41, 42 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wceq 1316 wcel 1465 wral 2393 crab 2397 wss 3041 c0 3333 csn 3497 word 4254 con0 4255 csuc 4257 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: oawordriexmid 6334 |
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